150 





we have successively 

 (1),,... 0=-^.«u+ ^•«'M+rf^:rfi- 



72,.' 



d-i/ d\ 



(2)m... 0= __^.«,,,+ ^-^ 



dx d'af . £^==0;' , d'x' . , rf^x' 



rfx' . d'x' - d'£ , d^x' 



(1)3.3 ... = — . «3.3 + -^, . «'3,3 + ^^;^. tU... + ^ . ^lA-r02., 



dy dW d'y' 



(2)3.3 ... = -rfjT • ws^ + ^j^2r • *"'•' +-7b^- '"'•'■'"'^' 



the two last of which equations reduce themselves, by means of the two first, to the following 

 form : 



«J.3 = "1,1- 'i«2fi> «'3,3 = ; 



a similar reduction gives in general, when < > 3, 



(l)t^ ... Ut^= Ul,i. tlV_i,(_l , 



(2)/,( ... 0= ivt^ + ^.2.«u,,s. at_j, ,_„ 



the -sum being taken from « = 2, to s=t — 2 ; so that the four first constants ki,i, tci,, , ho^, 

 1102,2, being determined by the four equations (l)i,i, (2)i,i, (1)2,2. (2)2,2, all the succeeding con- 

 stants of the same kind, 1*3,3, «m> ••• **'3,3> ^4,4,... are given by the following general expres- 

 sions, which may be deduced from the formulae (l)/,(, (2)(,t, either by successive elimination, 

 or by the calculus of finite differences ; 



"2r-|-l, 2r+l = "'.l' **'«''.2^ 5 "2r-|-2, 27-1-2 = ^ i J 



r being any integer number >0, and [^]'', [0]-'^, being known factorial symbols. In a simi- 

 lar manner we might calculate general expressions for the other constants of the form «/,<<, tuij,; 

 but it seems preferable to employ the following method, founded on the properties of partial 

 differentials, and on the development of functions into series. 



