An Elementary Treatise on Spherical Harmonics and Subjects Connected with Them |
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Page 3
... limits the name of Laplace's Coefficients to Zonal Harmonics , and designates all other spherical harmonics by the name of Laplace's Functions . The Zonal Harmonic in the case which we shall consider in the following chapter , i.e. , in ...
... limits the name of Laplace's Coefficients to Zonal Harmonics , and designates all other spherical harmonics by the name of Laplace's Functions . The Zonal Harmonic in the case which we shall consider in the following chapter , i.e. , in ...
Page 17
... ( P2 d − P , Ti ) + ( i − m ) ( i + m + 1 ) [ P , P_dμ = 0 , - -P άμ since the second term vanishes identically . F. H. 2 21 Hence , taking the integral between the limits - 1 ZONAL HARMONICS . 17 Expression of any function of μ.
... ( P2 d − P , Ti ) + ( i − m ) ( i + m + 1 ) [ P , P_dμ = 0 , - -P άμ since the second term vanishes identically . F. H. 2 21 Hence , taking the integral between the limits - 1 ZONAL HARMONICS . 17 Expression of any function of μ.
Page 18
Norman Macleod Ferrers. Hence , taking the integral between the limits - 1 and +1 , we remark that the factor 1 - μ2 vanishes at both limits , and therefore , except when im , or i + m + 1 = 0 , [ P.Pd , -0 . -1 = remark also that we ...
Norman Macleod Ferrers. Hence , taking the integral between the limits - 1 and +1 , we remark that the factor 1 - μ2 vanishes at both limits , and therefore , except when im , or i + m + 1 = 0 , [ P.Pd , -0 . -1 = remark also that we ...
Page 25
... limits are equal and opposite . Hence , ( m + i being even ) 1 m 1 [ __ " P ,. dp = 2 [ ' μ " P. dp μη S μm m and then Su " P. du = = 0 , if mi - 2 , i - 4 ...... μ We may proceed to investigate the value of S 1 μm Pi · αμ , if m have ...
... limits are equal and opposite . Hence , ( m + i being even ) 1 m 1 [ __ " P ,. dp = 2 [ ' μ " P. dp μη S μm m and then Su " P. du = = 0 , if mi - 2 , i - 4 ...... μ We may proceed to investigate the value of S 1 μm Pi · αμ , if m have ...
Page 26
... limits -1 and 1 , all the terms on the right - hand side will disappear except C , P2 , du , which will 2 become equal to C1 . 2i + 1 2i + 1 1 Hence C1 = μι Ρ . αμ , 2 -1 which is equal to 0 , if m + i be odd . Hence no terms appear ...
... limits -1 and 1 , all the terms on the right - hand side will disappear except C , P2 , du , which will 2 become equal to C1 . 2i + 1 2i + 1 1 Hence C1 = μι Ρ . αμ , 2 -1 which is equal to 0 , if m + i be odd . Hence no terms appear ...
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A. S. WALPOLE A. W. VERRALL Assistant Master Assistant-Master axis BEGINNERS BOOK Cambridge centre Classical Clifton College coefficients cos² Crown 8vo degree denoted density Edited by E. S. Edited by Rev ELEMENTARY English Exercises expression external point Extra fcap Fcap Fellow of St Fellow of Trinity Globe 8vo GRAMMAR Hence Illustrations internal Introduction and Notes J. P. MAHAFFY JOHN BOND John's College late Fellow LATIN Lecturer Litt LL.D numerous obtain Owens College Oxford P₁ P₂ plane potential preparation PRIMER radius rational integral function revised and enlarged satisfies the equation School Second Edition sin² solid solid harmonic sphere spherical harmonic suppose surface harmonic Tesseral Translated TREATISE Trinity College University V₁ Vocabulary y+b² y+c² zonal harmonics αμ αμσ µ² µ³
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