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By Ferrers, N. M. (Norman Macleod)

This quantity is made out of electronic pictures from the Cornell college Library ancient arithmetic Monographs assortment.

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Additional resources for An elementary treatise on spherical harmonics and subjects connected with them

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U f? . 9;6 P 2< I _1_ ~ / "'I (f)'\ v-' y* For each of these expressions, when substituted for V, 2 the equation V V = 0, and they become respectively is put = 0, and equal to (1) and (2) when consequently satisfies r = z. ') becomes equal to (2') when r = c, and will great, therefore denote the required potential for all values of r less than c. These expressions means may be reduced to other forms by of the expressions investigated in Chap. 2, Art. 25, viz. Or P. 3 APPLICATION OF ZONAL HARMONICS 46 which is equivalent to M The brings it substitution of the last form of into the form 1 fa c*_ 2t{2 + (2?

Equal to r while o ^ increases from to TT, be greater than cannot become ^ and therefore the expression under the integral sign cannot become infinite. Supposing then that we write z for we a, and V 1 p for 6, get 1 /> d*t z We may remark, and is V 2 (a + p^ in passing, that d* r Jo z 1 _ - V~l p cos ^ ^ =r I/a cos^- Jo z + \f I therefore wholly real. Supposing that 8 /j = ic2 + 2 2/ i , and that x*+y*-{- z thus obtain d^ 1 = r9 we , 40 ZONAL HAEMONICS. A^ivi times with respect to i _<* Hence this, * IT Again, for and S-, 2 m ' /r)* r for p, (1 yuA for a, _ a __ ___ / a TT J 2 6 2 1 r* b cos and we get ' ^ and + (^ becomes 1 1)^ up 2/A to ^ h for 6, (/J?

4... 19. be even, and . 4... (i cos 6, if i be odd. 1) Let us next proceed to investigate the value of Clt P cos m0 sin t Jo 6 1 d0. ZONAL HARMONICS. SO Tins might be done, by direct integration, from the above Or we may proceed as follows. The above value of P when multiplied by expression. i is (that by ~ (sin Zt (m+ 1) 6 sin cos mO sin 1) 9}} will consist of (m a form {i 2n + (m I)} 0, that odd multiples of 0, as i + m is odd or even. and TT it Therefore, when integrated between the limits We may therefore limit ourwill vanish, if i + m be odd.

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