153 



and if we change the function w to x and i^ successively, we find the following developments for 

 those two variables, according to the powers and products of », »' ; 



X = 2 1 oi-\ ro 1-'". »"»'"'. — f— —^L. 1 



zivj .Luj .«* -df-Uf^-^yvt' dt )' 



in which Z, Z are formed from z, 2', by changing x to t, and x' to Vt'. Applying these results 

 to the equations (H(S)), which are of the form 



C=z + «/ ,= V(2' + »'<?); 

 we find the following expressions for ^, u, as functions of 2 and i, 



I') 



f = 2 [0]-". rO]-"'. 2". ^ — r 



K= 2.ro"!-". roi-«'. 2»'. " 



(MW) 



in which 



a"' d.f f" dd" 



^ 2Vz' dVz' • ^ 2^z' dz ' '^^ ' 



f, (p, being deduced from the formulae (K(5)) by changing ^ to z, and » to ^z', and in which we 

 may make after the differentiations a' = l. And differentiating these developments (MW) in 

 the manner already prescribed, we find, finally, the following general expressions for the con- 

 stants tt(^/, toi^t' ; 



M+ t'+n+ ,u—2 nn,ni) 



"'." = tOJ-'- 1«]-'' ^- [O]-'- CO]-"'. 2"'. ^,,^^.^„-,,^.,. 



-,,= [0]-. CO]- 2. [0]-. [0]-"'. 2- ;,,,,,^„_,,^ 



n,n', being any positive integers, and («, z, 2') being changed after the differentiations to 

 (0, 0, 1). It may be useful to observe, that by the formulae (Nt^)), and by the nature of the 

 developments, we are to make 



^-2.^(0,0) _ rf-s.„(0,o; _ ^ 



rf2-l.rf2'-l~^' rf2-».rf2'-» " "' 



These expressions (OC^)), may be put under other forms, some of which are more convenient 



