509 



particle d/j.. We have thus the means of expressing the second line 

 in the proper form ; and if we write 



2 (au + a'v 4- a"w) dfj.= 



2 



then the required expression of the second line will be 



which, if we put 



K=i (A0' 2 +B0' 2 + . . 



may be more simply represented by 



Only it is to be remarked that A, B, . . H, . . P, Q, . . will in general 

 contain not only 6, (f>, . ., but also the differential coefficients 6', 



$',.., and that in forming the differential coefficients . t , &c., 



dti dfy 



the quantities 0', 9', . . , in so far as they enter into K, not explicitly, 

 but through the coefficients A, &c., must be considered as exempt 

 from differentiation, so that the preceding expression for the second 

 line by means of the function K is rather a conventional representa- 

 tion than an actual analytical value. 



Uniting the two lines, and equating to zero, the coefficients of 

 20, cty, &c., we obtain finally the equations of motion in the form 



d_dT_dT dV dK = 

 dt dO' dd dd dd' 



^<fT_rfT </V rfK 



dt d$ d$ d<j> dtf ~ '. 



where the several symbols are to be taken in the significations before 

 explained. 



2 p 2 



