18 Proceedings of the Royal Irish Aeao^emy. 



no virtual moments when the 6's are varied, but the virtual moment, 

 when 01 is varied, is $iS(/)i, and when (^2 is varied •I'aS^a, &c. Therefore 



5-7-5^1+5 — 502 + &c. + 5 -— ■ 5^1 + (S:c. = (0i59i + + (0„+i + *i) Sd)i 1 &c.) 5^ 



Therefore, 



dT dT 



5 — = 0i5<, S— r = 025<, &c., 



dQi ddz 



s -T- = (0„u + *!) 5z;, 5 -^ = (0„+2 + *2) 5i;, &c. 



T is of the form 



2r=au0i^+ +«n*i,n^T<?'i'^+ +20.1^6182+ +2ai,„.iei^i 

 + 25i0i + + 25„+i(^i + + c, 



where the a^, ai2 ■ ■ ■ ii, i-z • ■ • c are functions of the co-ordinates and 

 the time. Therefore 



Therefore 



dT . . . , 



— ^ = ffll^l + «1292 + + «! . 7i+l<?l + + 5l. 



rf01 



(^T • ■ . • 



5 — - = ffii50i 4 ^12502 + «i . jifi 5(^1 + 0i5aii + +5Ji. 



^9i 



Since the body is at rest just before motion, 6*1, &c., are of the first 

 order of small quantities. Scf^i, 8^2 ■ • • are zero, since ^i, ^j, &c., 

 are the co-ordinates of constraint. S^i is of the same order of small 

 quantities as {Sty, and, consequently, we may neglect aU terms having 

 multipliers of the type 80. 

 Again, 



Therefore 

 consequently. 



dbi dbi , db 



ddi ddi di 



dby 

 Sbi = — 8t; 

 at 



dT . . .dbi 



S -^ = aii59i + ai259a + amSOn H St, 



ddi dt 



dT . . dhi 



I -jz- = anBdi + ann Bd,, + -r- St, 



adz dt 



dT . . dbn^i 



S -rT-=ai.n*l 501 + rt„, „,1 SSn + -j— St, 



a(p\ at 



&c., &c. 



