12 



Connexions between the Partial Differential Coefficients of the two 

 Characteristic Functions. 



5. The connexions between these coefficients, are to be obtained 

 by differentiating the preceding expressions for 



3F IV W iW iW IW 

 3x ' 3y ' 32 ' IT' 3^ ' 3y ' 



and by attending to the homogeneous forms which we have assigned 

 to V and W. The dimension of W being supposed = 0, we have by 

 the nature of homogeneous functions, 



»-5-+''-^^ +y-j— =0; 



3« 



3^ 



h 





iw 



+ « 



3y 





+ /3 



3/3* 



+ y 



w 



3/33y 



(F) 





&c. 



We have also, by the homogeneous nature of v^ which we have 

 put under the form of a function of the first dimension, the following 

 relations : 





3« 



3u 



+ ^ 3T + ^ i; = '' 



3/3 • ey 



¥v , „ '^v , 3^^ ~. 



* 3?+ ^-§;5^ +y -u;'^^' 





3^^ 



3/33y 



3'i> 



0£Coy 



3/3Sy 3y^ 







Set'Sy 



&c. 



(G) 



