346 ON THE MOTION OF 



These last formulas, which are well known to mathemati- 

 cians, will enable us to find the values of L, L\ L", L„ M„ N„ 

 in all cases where the surfaces are known, and thereby to put 

 their differential or variational equations in the forms above 

 given (18); forms which will always be found remarkably 

 well adapted to geometrical and mechanical inquiries, from 

 the facility with which the analytical results can be translated 

 into the language of geometry. Between these cosines there 

 exist the following relations : (20) 



L = aL-hbM, + c N, , 

 L' = a'L.-hb'M^e'N,, 

 L = a"L,+ b"M, + c"N l i 



L = a L-ha'L'-ha"L", 

 M t = b L+b'L'+b"L\ 

 N, = cL+c'L 1 +c"L". 



Taking now the variations of equations (2), and recollecting 

 that x„ y, z, are no longer constant, we obtain (21) 



dx = dZ -+-a dx,H-6 ty,4-c bz^xfla -\-y,bb -+-z,dc , 



dx> — df -^a'dx.-i-b'dy^c'dz^x^a' -hy,db' -^zfic' , 

 i z » — dZ"- h a"dx l -hb"dy l -hc"dz i + z l da" + y l db" + z l dc". 



Adding these equations together, after multiplying the first 

 by L, the second by L', and the third by L", and then redu- 

 cing by means of the formulas (20), there results 



— (L dx -+- L'dx'-h L"hx") -+- x f (Lda -+- L'M+LW) ) 



+lLdZ+L'dZ , + L' , d£ , j + y,(Ldb-)-L / db , -t-L'<db'')\=o. 



^.(Lfa+Mfiy, H- N,dz, ) -f- z,(Ldc -+- L'dc'+ L"8c ' ) ) 

 Substituting in place of the variations of the nine cosines their 



