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6. The first process shews, by the aid of Lagrange's
Let y denote a quantity, such that
P1 = 2.1.2... i dμ3
7. From this form of P, it may be readily shewn that the values of μ, which satisfy the equation P= 0, are all real, and all lie between 1 and 1.
For the equation
(μ2 - 1) = 0 has i roots = 1, and i roots = — 1,
(u2 - 1)'=0 has i-1 roots = 1, (i-1) roots=-1, and one root = 0,
(-1)=0 has (i-2) roots = 1, one root between 1 and 0, one between 0 and -1, and (-2) roots = −1, and so on. Hence it follows that
It is hardly necessary to observe that the positive roots of each of these equations are severally equal in absolute magnitude to the negative roots.
8. We may take this opportunity of introducing an important theorem, due to Rodrigues, properly belonging to the Differential Calculus, but which is of great use in this subject.
The theorem in question is as follows:
If m be any integer less than i,
It may be proved in the following manner.
If (x-1)' be differentiated i-m times, then, since the equation
(x2 — 1)' = 0
has i roots each equal to 1, and i roots each equal =-1, it follows that the equation
has (im) roots (i. e. m) roots each =1, and m roots each = 1, in other words that (x-1) is a factor of
dx2 ̄m (x2 — 1)o.
We proceed to calculate the other factor.
For this purpose consider the expression
(x + α ̧) (x + α2) ... (x + α1) (x + B1) (x + B2) (x + B1).
Conceive this differentiated (I) im times, (II) + m times. The two expressions thus obtained will consist of an equal number of terms, and to any term in (I) will correspond one term in (II), such that their product will be (x+α1) (x + α2) ... (x + α ̧) (x + B1) (x + ẞ2) ..... (x + B1), i.e. the term in (II) is the product of all the factors omitted from the corresponding term in (I) and of those factors only. Two such terms may be said to be complementary to each other.
Now, conceive a term in (II) the product of p factors of the form x+a, say x+a', x + ά' ...x + a(2), and of factors of the form +B, say x+B, x+B,,... x + B12) Bur We must have p+q=i-m.
The complementary term in (I) will involve
p factors x+B, x + B" ... x + BIP),
Now, every term in (I) is of i+m dimensions. We have accounted for p + q (or i-m) factors in the particular term we are considering. There remain therefore 2m factors to be accounted for. None of the letters
There will be another term in (II) containing (x + B′) (x + ß'') ... (x + B(1) (x + α) (x + a,,) The corresponding term in (I) will be, as shewn above, (x+α) (x+a') ... (x + a(1)) (x+ẞ,) (x + B) ... (x + B1) (x+,α) (x+,2)... (x + m2) (x +‚ß) (x + „B) ... (x + „B).
Hence, the sum of these two terms of (I) divided by the sum of the complementary two terms of (II) is
(x + ̧x) (x+ ̧α) ... (x + m2) (x + ̧B) (x + ‚ß) ..... (x +mB).
Now, let each of the a's be equal to 1, and each of the B's equal to -1, then this becomes (x2 - 1)". The same factor enters into every such pair of the terms of (I). Hence
The factor may easily be calculated, by considering that
the coefficient of x+ in i+m
dim (x2 - 1);
and that the coefficient of x3-m in
is 2i (2i−1)...(i+m+1),
2i (2i − 1) ... (i + m + 1) (i + m) ... (i − m + 1). .
9. This theorem affords a direct proof that C- (u2-1),
C being any constant, is a value of ƒ (u) which satisfies the equation
Hence, the given differential equation is satisfied by put
Introducing the condition that P, is that value of ƒ (u). which is equal to 1, when μ= 1, we get
10. We shall now establish two very important properties of the function P; and apply them to obtain the development of P in a series.