By Henry Thomas Herbert Piaggio

ISBN-10: 1177592886

ISBN-13: 9781177592888

Initially released in 1920. This quantity from the Cornell collage Library's print collections used to be scanned on an APT BookScan and switched over to JPG 2000 layout via Kirtas applied sciences. All titles scanned hide to hide and pages may possibly comprise marks notations and different marginalia found in the unique quantity.

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**Extra info for An elementary treatise on differential equations and their applications**

**Sample text**

Pj 3 (2t- 1) 9) P^ P^. rift this equation we deduce and 1 being taken, in order that at the superior limit. P f P ( _2 may ZONAL HAKMONICS. 38 to the fundamental equation for a zonal see that Now, recurring harmonic, we 1 24. We have already seen that r P Pm dp = 0, I t i and ra 1 Suppose now that being different positive integers. required to find the value of JI n We have P Pm t it is d/j,. already seen (Art. 10) that And, from above, ' i+1 -P V _ p . ZONAL HARMONICS. 39 "We "will next proceed to give two modes of expressing Zonal Harmonics, by means of Definite Integrals.

F- + ... + -^^- + ... = o, last ZONAL HARMONICS. 22 X o* /y the /v ^f-2 i number of equations, i> forms x and the a, number and ^ ^j We 1 if ^V i and therefore of 4~ 1 i be odd, = if i being 2> 1 if i be even 2i of letters of the form a being i i\ be odd, if i 2 obtain, as before, *-. ^, t) (ca + ^, and then putting 6 - Gi _ 2) (a, (i-2 , To apply + e) . . (to - a.. J 4 2 . . a^, . . (a4 _ 2 +&>)... _ 2J ~ ^) (a^ - a _ ) = + &>)... _ a = = * - 2* ... 2 _ = i-2s- 1... i _ 2t i 2, Hence .

To - a.. J 4 2 . . a^, . . (a4 _ 2 +&>)... _ 2J ~ ^) (a^ - a _ ) = + &>)... _ a = = * - 2* ... 2 _ = i-2s- 1... i _ 2t i 2, Hence . ^i-8, and ; be even. ^ 15. 9 - 1) (2i - 2g - 3). {2 (^ - 2g) - 1|... {(- 2s - 1) or (t - 2s) ( | ^ } ' aj we see, for A, ZONAL HARMONICS. 4 We give the values of the several zonal harmonics, P inclusive, calculated by this formula, to 10 P_~ 8 ^ 2 "~ 1 2 2 2 ' ' from 24 ZONAL HARMONICS. 1 . 4^ 8 _ 231/I. - 315//, + lOo/i - 5 ~~ 4 p _ 13. 11. 4x2^ + 2x 2. 4^ 15. 8 ^ .

### An elementary treatise on differential equations and their applications by Henry Thomas Herbert Piaggio

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