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x dx


= 0 :

(3) 6 co ^ (1) + (3) + (2) = 0 gives, on equating to zero the coefficients of each differential,


y' + hl + 1 = 0, λ

tu + m = 0, a a

02 12

+ n = 0.


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Multiply by X, Y, ã, and add, then by the equations of condition,

1 + 1 + 8 = 0. Substituting for 1 in the preceding equations they become M (0-x')= a’l-8x, P (7-4)=b'm-dy, M (z-z')=con-87, whence


a'l- bom

on – da

= p suppose. Now multiplying numerator and denominator of these


y fractions by

respectively, and adding together

12 the numerators and the denominators,

y'? +

62 Р:

lx' + my nz But on multiplying the numerator and denominator of these fractions by l, m, n respectively, and adding the numerators and the denominators, we also have

8 - (lx' + my' + nz') Р

a'l + b'm + en - 8 Therefore equating the two values of р


{(12 + m 2 + x )* a? 62 c?

al + m + en -34 as the required equation to the surface.

(16) Find the equation to the surface which is constantly touched by the plane

lr + my + n z = V,

we have


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1, m, n, v being connected by the equations

[ + m2 + n° = 1,

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Differentiating with respect to l, m, n, v we have,

(1) wdl + ydm + xdn = dv.
(2) ldl + mdm + ndn = 0.

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= v dv

(v2 – a ́) (o? – 62) ^ (1) = u (2) + (3) gives, on equating the coefficients of each differential,

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(6) 18 = un +

v? - ?

(7) 1 = v

+ (v– a’)2 * (v* – 6?) * (o - c)?] 1(4) + m (5) + n (6) gives by the conditions,

(8) λυ = μ,
X (4) + y (5) + < (6) gives

a po? = ци+
v2 a v2








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whence (10) X? (19 – 1) = by (7) and (3);


and n =

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and therefore X=

v (pu2 – v“)' Substituting these values in (1) we have

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go? - c v2 ci
Multiply by x, y, z and add, then by (9) and (10)

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This is the equation to the surface of a wave of light propagated through a crystalline medium. See Fresnel, Mémoires de l'Institut, Vol. vii. p. 136; Ampère, Annales de Chimie et de Physique, Vol. xxxix. p. 113; and Smith, Cambridge Transactions, Vol. vi. p. 85. If from the above equation we subtract

za + y +


go2 and reduce, we find


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which is the form of the equation given by Fresnel.



In this chapter I shall collect those Theorems in the Differential Calculus which, depending only on the laws of combination of the symbols of differentiation, and not on the functions which are operated on by these symbols, may be proved by the method of the separation of the symbols: but as the principles of this method have not as yet found a place in the elementary works on the Calculus, I shall first state briefly the theory on which it is founded.

There are a number of theorems in ordinary algebra, which, though apparently proved to be true only for symbols representing numbers, admit of a much more extended application. Such theorems depend only on the laws of combination to which the symbols are subject, and are therefore true for all symbols, whatever their nature may be, which are subject to the same laws of combination. The laws with which we have here concern are few in number, and may be stated in the following manner. Let a, b represent two operations, u, v two subjects on which they operate, then the laws are

(1) ab (u) = ba (u),
(2) a (u + v) = a (u) + a (v),
(3) a.a". u = a

ar+.u. The first of these laws is called the commutative law, and symbols which are subject to it are called commutative symbols. The second law is called distributive, and the symbols subject to it distributive symbols. The third law is not so much a law of combination of the operation denoted by a, but rather of the operation performed on a, which is indicated by the index affixed to a. It may be conveniently called the law of repetition, since the most obvious and important case of it is that in which m and n are integers, and am therefore indicates the repetition m times of the operation a. That these are the laws employed in the demonstration of the principal theorems in Algebra, a slight examination of the processes will easily shew; but they are not confined to symbols of numbers; they apply also to the symbol used to denote differentiation. For if u be a function of two variables w and y, we have by known theorems in the Differential Calculus,

d d

d d (u)


dy dv
Also considering u and » as functions of x only,


(u + 7) (u) + (0),

dx dy




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The principal theorems in Algebra which depend on these laws, and which have therefore analogues in the Differential Calculus, are the Binomial Theorem with the great number of theorems— Exponential, Logarithmic, and others—which are derived from it; and the theorem of the decomposition of a multinomial of any order into simple factors with the various consequences which are deduced from it.

It is to be observed that in all the applications of this method to the Differential Calculus, a constant has the same laws of combination with the differentials that they have with each other, and therefore the theorems are true for complex symbols involving constants and symbols of differentiation. Also, there are two ways in which symbols of differentiation may differ from each other, either by having reference to different variables in the same function, or by having reference to different functions of the same variable, and this difference gives rise to two totally distinct series of theorems, as will be seen in the following examples.

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