SIMILAR TO THOSE OF HYDRODYNAMICS. 385 



with similar expressions resulting from the projection of ds 2 , ds 3 , .... ds n on the 

 axes. Comparing these equations with (39) and writing 



U+q=W (99) 



gives 



j t (Mn + M12 + + Mi») T = dW ( 10 °) 



and by (88) 



j t (uiA u + u 2 A 12 + .... + u n A ln ) = J -^ (101) 



and in like manner 



J t («l4.1 + M»2 + .... + UnAm) = Jdf n 



Assume a function P such that 



then since 



W=*§ (102) 



we have 



An ^ + A m ^ + . . . . + 4. ^ = J^ etc. (103) 



(dP ) 



u x A n + n. 2 A 12 + + t« R 4 lre = J j ^ + i//! j 



«1^21 + Ma4a + ...-+ M„An — J \ rf^ + ^2 | (104) 



Mm + M»2 +-... + U n A lm = J U^ + ijr„ | 



The functions i/» contain U 6 2 , . . . . #„ but not t. Solution of (104) gives 

 _ M x UP \ d0 t UP \ d6 n ( dP ) 



M i- S" \ m, + h\ + 7^ \de a + *{ + ■ ■ • • + ^ \W n + *•! etc - 



bearing in mind that 



II A, II = J"' 1-1 (105) 



These equations for ^, ^ . . . . u n may be written in the form 



