152 



F=i\f. *=:«^(p, (Its)) 



and therefore 



rf6« r -r L J L J da^-db"" ^ < 



rfj/ d'x' d'"+"''j^ 



(KW) 



m, m', being any positive integers which satisfy the following relation 



2 m + »»' > 2. (LW) 



If then we eliminate^, d, between the equations (HC^)), so as to find expressions for those va- 

 riables as functions of r and 6 ; it will remain to differentiate those expressions, (i) times for 6, 

 and {V) times for z, and to put after the differentiations tf = 0, 2=0; since the partial differ- 

 entials thus obtained, multiplied by the factorial quantity [0]"*. [0]"", will give, by (F(^>), the 

 general expressions that we are in search of, for the constants Ut^i, to<,(/. 



To perform this elimination, we may employ the theorems which Laplace has given, in the 

 second book of the Mecanique Celeste, for the developrnent of functions into series. Laplace 

 has there shewn that if we have any number (r) of equations of the form 



xzz^{t+ ctz), x' = ^^(i' + «'2'), x'i=U[t" + u."z"), &c. 



in which z, z', z", &c. are functions of x, x", x", &c. and if we develope any other function u of 

 the same variables, according to the powers and products of «, »', x", &c. in a series of whicii 

 the general term is represented by qn,n/,nii, ... »"•»""•*"''"• &c. ; we shall have to determine the 

 coefficient gn.m.nu^..., a formula which may be thus written, 



ffn+n'+B//-)-...— r . Jr^ \ 



qn,n,,.,,... = [0]-". [0]-'. [0]-" ... at„-lJi,n,ZTji„r.„-_l [dZdZtjC J 



u, being a function formed by changing in u the original variables x, x', x", ... into other varia- 

 bles determined by the following equations 



ar=ip(< + «z"), x' = -4^(t' + a'z'"'), x" = n(<" +•"«"""), &c. 



and «, a', «", ... being supposed to vanish after the differentiations. Laplace has also shewn, 



that when there are but two variables j;, x', the partial differential ( - — ; — fiT" )> determined in 



\ax.d» .d» ..■/ 



this manner, reduces itself to 



{^) = ---■ m + '■'■ m (t) +-■ ("^)- &) ■ 



in which Z, Z,' u, represent the values that z,' z', u, take, when wc suppose « = 0, «' = 0. If 

 then the original equations are of the form 



