606 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



^ /m+p- 

 _ / i\-(v+i) sinmTT /m + 1 . a-'+P 



&c. = &c. 



sinm+p + lir d-'-P-*'^^ z"' 



-^^D^^n^i ^ ^^_^^^^^ sm(m-l)7r _^^^^-+p_ 

 dz-^P*'^^ sin (m+p) IT /m+p + 1 



sinmTT /m + 1 



sm(m+p + l)'7r mlm+p + 1 



P = 2A/^TT|(-1)>'+1 — 



sinmir rf^-^*"*^^ 



+ !L . (_l)-(P+i) sin(m+/^ + l)7r ^ ^^^^' . ^"^^ 

 1 sin wTT rf^-(^+'' 



+ ^r-7>'<-^) ■ ^-^ — m(in-l) 



1.2 sin w TT ^ ^ 





rf^^-P + 1^ 



or 



+ ... \ 



1 i. . Ji ) 



. ) T / -.N.j-i sin(w + o + l)7r </-(p+i) , 



^ Bin mir dz-^*^' ^ ' 



/ -iN„*i / T sin(»i+jo + l)'7r rf-^^+i' 



= (-1)""' IP + 1 . COS (;> + 1) TT ^^,-71^ • (^ + « 



if ?» be a whole number, and p a fraction or whole number. 

 Ex. Let P = X (p{d + a) = e + a 



d 6 {Q + a) {z—6) = — + —- by integration 

 — - (« + «)= -jr— 5 + — —- = the same as the other. 



dz-^ ' ' 2.3 



It must be observed that the integrations are performed separately, as in the 

 demonstration. The problem which led me to this theorem is that of finding 

 the law of force by which the particles of a sphere must act on a point, so that 

 the whole attraction may be the same as though the sphere were collected at its 

 centre of gravity. Unfortunately the question leads to a general differential 

 equation, the solution of which we have not as yet been able to effect. Still we 

 have done all that is requisite in order to exemplify the use of our analysis, by 

 shewing that this question reduces itself to such an equation : we shall, therefore, 

 exhibit our process. 



