An Elementary Treatise on Spherical Harmonics and Subjects Connected with Them |
From inside the book
Results 1-5 of 18
Page 9
... proceed as follows : If p , be the coefficient of h ' in the expansion of H , H = 1 + ph + ph + ... + ph + . .. hH = h + p ̧h2 + P ̧h3 + + p ̧hi + 1 + ... ... d2 : . h dh2 ( hH ) = 1.2p , h + 2.3p , h2 + + i ( i + 1 ) p ; h ' + ...
... proceed as follows : If p , be the coefficient of h ' in the expansion of H , H = 1 + ph + ph + ... + ph + . .. hH = h + p ̧h2 + P ̧h3 + + p ̧hi + 1 + ... ... d2 : . h dh2 ( hH ) = 1.2p , h + 2.3p , h2 + + i ( i + 1 ) p ; h ' + ...
Page 11
... in the result putting = μ . Both these methods how- r ever would be somewhat laborious ; we proceed therefore to investigate more convenient expressions . 6. The first process shews , by the aid of ZONAL HARMONICS . 11 PAGE.
... in the result putting = μ . Both these methods how- r ever would be somewhat laborious ; we proceed therefore to investigate more convenient expressions . 6. The first process shews , by the aid of ZONAL HARMONICS . 11 PAGE.
Page 14
... proceed to calculate the other factor . For this purpose consider the expression ( x + α ̧ ) ( x + α2 ) ... ( x + α1 ) ( x + ẞ ̧ ) ( x + B2 ) ... ( x + ß . ) . Conceive this differentiated ( I ) im times , ( II ) i + m times . The two ...
... proceed to calculate the other factor . For this purpose consider the expression ( x + α ̧ ) ( x + α2 ) ... ( x + α1 ) ( x + ẞ ̧ ) ( x + B2 ) ... ( x + ß . ) . Conceive this differentiated ( I ) im times , ( II ) i + m times . The two ...
Page 25
... proceed to investigate the value of if m have any other value . For this purpose , resuming the notation of the equations of Art . 13 , we see that , putting 0 = m + 1 , and ∞ = ∞ , we have X1-2 a , + m + 1 a1-2 + m + 1 = X1 Xi - 28 + ...
... proceed to investigate the value of if m have any other value . For this purpose , resuming the notation of the equations of Art . 13 , we see that , putting 0 = m + 1 , and ∞ = ∞ , we have X1-2 a , + m + 1 a1-2 + m + 1 = X1 Xi - 28 + ...
Page 26
... proceed to shew . Suppose that m is a positive integer , and that μm is de- veloped in such a series , the coefficient of P , being C , so that μm = = ΣC , P .; i then , multiplying both sides of this equation by P , and inte- grating ...
... proceed to shew . Suppose that m is a positive integer , and that μm is de- veloped in such a series , the coefficient of P , being C , so that μm = = ΣC , P .; i then , multiplying both sides of this equation by P , and inte- grating ...
Other editions - View all
Common terms and phrases
Assistant-Master axis b²+w become infinite bounding surface C₁ Cambridge centre Chap co-ordinates coefficients component attraction confocal ellipsoid corresponding cos² Crown 8vo degree denoted distance dV dV dy dz Edited by Rev ellipsoidal harmonics equal Eton College expression external point Extra fcap factor fcap Fellow of St follows Greek Hence homogeneous function internal John's College lamina late Fellow LATIN monics multiplying numerous obtain Oxford P₁ P₂ plane positive integer potential Professor radius rational integral function satisfies the equation series of zonal shewn sin² solid angle solid harmonic solutions sphere spherical harmonic spherical shell suppose surface harmonic Tesseral thickness Trinity College V₁ writing y+b² y+c² zonal harmonics αμ αμσ µ² µ³ µ³)³ προ
Popular passages
Page 17 - THE SEVEN KINGS OF ROME. An Easy Narrative, abridged from the First Book of Livy by the omission of Difficult Passages; being a First Latin Reading Book, with Grammatical Notes and Vocabulary.
Page 52 - HALES— LONGER ENGLISH POEMS, with Notes, Philological and Explanatory, and an Introduction on the Teaching of English. Chiefly for Use in Schools. Edited by JW HALES, MA, Professor of English Literature at King's College, London.
Page 21 - AN ELEMENTARY TREATISE ON THE LUNAR THEORY, with a Brief Sketch of the Problem up to the time of Newton. Second Edition, revised. Crown 8vo. $s. 6d. HEMMING— AN ELEMENTARY TREATISE ON THE DIFFERENTIAL AND INTEGRAL CALCULUS, for the Use of Colleges and Schools.
Page 18 - Astronomer Royal :— ELEMENTARY TREATISE ON PARTIAL DIFFERENTIAL EQUATIONS. Designed for the Use of Students in the Universities. With Diagrams. Second Edition. Crown 8vo. $s. 6d. ON THE ALGEBRAICAL AND NUMERICAL THEORY OF ERRORS OF OBSERVATIONS AND THE COMBINATION OF OBSERVATIONS.
Page 14 - MAYOR (JOSEPH B.)— GREEK FOR BEGINNERS. By the Rev. JB MAYOR, MA, Professor of Classical Literature in King's College, London. Part I., with Vocabulary, is.
Page 2 - IV. and V. Edited by CE GRAVES, MA, Classical Lecturer, and late Fellow of .St. John's College, Cambridge.
Page 18 - Crown 8vo. 6s. 6d. ON SOUND AND ATMOSPHERIC VIBRATIONS. With the Mathematical Elements of Music. Designed for the Use of Students of the University.
Page 25 - ARITHMETIC. (Dedicated to Lord Sandon.) With Answers and Hints. Standards I. and II. in box, is. Standards III., IV., and V., in boxes, is. each. Standard VI. in Two Parts, in boxes, is. each. A and B papers, of nearly the same difficulty, are given so as to prevent copying, and the colours of the A and B papers differ in each Standard, and from those of every other Standard, so that a master or mistress can see at a glance whether the children have the proper papers.
Page 22 - JACKSON — GEOMETRICAL CONIC SECTIONS. An Elementary Treatise in which the Conic Sections are defined as the Plane Sections of a Cone, and treated by the Method of Projection. By J. STUART JACKSON, MA, late Fellow of Gonville and Caius College, Cambridge.