292 



PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



> . (129,) 



.3 = (I, - «,)2 + fe _ p^)2 + (^2 _ y^)2, 

 S, = 1,2 ^. ,^2 + ^^2^ ^^ = (1^ _ 1^)2 + (^^ ^ ^^)2 + (^^ _ ^^)2^ 

 d, = a^^ + ^^2 + y^2^ ^^ = (1^ _ ^^)2 + (,^ _ ^^)2 + (^^ _ y^)2^ 



ds = (ll - «2)' + (^l - N' + (^1 - 72)', 

 d, = l,^ + ;?2=^ + ^2^, d, = (li - «i)2 + (^j - p^)2 + (^^ _ y^)2. 



These ten distances Vs^, &c., are not, however, all independent, but are connected by 

 one equation of condition, namely. 



0=*i2 



* 2 5 2 ^ o 2 e 2 _|_ c 2 ^2 ^ o 2 .2 ^ ^2 g 



2c2 



+ *l2 d.2 + ^,2 J^2 + ^^2 d2 + ^^2 ^^2 + ^,2 ^^2 

 + ^,2 d^2 + ^^2 ^^2 + d^2 d2 + ^^2 ^^2 + ^^2 ^^2 



— 2 ^1 ^3 *4 2 Jfg A'4 -^5 2 .S3 55*1 — 2 .^4 S1S2 — 2 55 .^2 *3 

 -2V'^3^3- 2522a4C?4- 2.^32 .ygC?^- 2 .$42 .y, C?, - 2 .¥52 ^^ ^^ 



— 2 .Si2 ^^ ^,^ — 2 .92^ 55 </4 — 2 .S32 .y, J5 — 2 .^42 s^d-^ — ^ s^ s.^ d^ 



— Is^d^ d} — ^s.^d^ d^ — Is-^d^ d^ — 2s^d^ d^^ — 2s^d^ d^ 



— 2 s^d^^ d^ - 2 s^d^^ d^ — 2 s^d^^ d^ - 2s^d^^ d^^ 2s^d^ d^ 



— 2 d^ d^ d^ — 2dc^ d^ d^ — 2d^ d^ d^~-2d^ d^ d^ — 2d-^d^ d^ 



— ^SyS.^s^d.^— ^s^s^s-^d^ - ^Sj^s^s^dr^ — \s{s^s.^d^ - 4s^s.^s^d2 



— 4 Si c?2 do^d^ — 4 S2 d^ d^d^ — 4 .S3 d^ d^d^ — 4 s^ d^ d^d^ — 4 s^ d^ flfg ^3 



— 2 5j 1V2 .S3 "4 — 2 .^2 "^3 "^4 "5 — 2 ^3 .S4 .S5 a J 2 .S4 s^ s^a2 — 2 Ss^ s^ $2 d^ 



— 2s^s^d]^d2 — 2 $2 s^ f/g ^3 ~ 2 .S3 .S5 d^d^ — 2 .S4 s^ d^d^ — 2 ^5 ^g ^5 ^1 



— 2 Si d^ d^d^ — 2 *2 ^2 ^4 ^1 — 2 .S3 6?3 <4 ^2 ~ 2 .S4 </4 c?i c?3 — 2 ^5 c?5 c?2 ^4 

 "T 2 .Sj .S2 .S3 .S4 + 2 .Sg *3 .S4 .S5 -j- 2 .S3 .S4 .S5 .Sj 4- 2 .S4 .S5 .s, .S2 + 2 .s^ .s, .Sg % 



+ 2 .Sj .S2 .S4 6/3 + 2 .S2 *3 s^d^ + 2 *3 .S4 .Sj J5 + 2 .S4 ^5 .S2 c?i + 2 .S5 .s^ .S3 Jg 

 + 2 .Sj .S3 .S4 c?i + 2 .S2 .S4 .S5 c?2 + 2 .S3 .S5 s^d^ + 2 .S4 .Si ;S2 fi?4 + 2 s-^ .Sg .S3 d^ 



+ 2 *i .S2 C?3 C?4 + 2 .S2 .S3 C?4 C?5 + 2 .S3 .S4 C?5 Jj + 2 .S4 ;S5 f/j ^2 + 2 «5 -^l ^2 ^3 



+ 2 .Sj .S3 c?2 ^3 + 2 .S2 .S4 <:?3 ^4 + 2 .S3 .S5 d^d^-\-2 s^ s^ d^di-\-2 s^ .Sg d^ <4 



4- 2 .Si .S4 «Zi ^2 + 2 .S2 .S5 C?2 ^3 + 2 *3 .Sj ^3 fi?4 + 2 .S4 .S2 ^4 C?5 + 2 .S5 .S3 £?5 rf, 



+ 2 *i .S4 £?, <Z3 + 2 .S2 .S5 ^2 ^4 + 2 .S3 Si d.^d^-\- 2 s^ .Sg d^di-\- 2 s^ s.^ d^ c?2 

 + 2 .Sj .S4 C?2 <^3 + 2 A'2 *5 ^3 ^4 + 2 .S3 .s, d^d^-\-2 A'4 ^2 d^ d^ -^ 2 S^ .S3 </i c?2 

 + 2 .Sj «4 C?3 C?4 + 2 .S2 .S5 <?4 C?5 + 2 .S3 ^j rfg C?, + 2 ^4 .Sg <?! 6?2 + 2 .S5 .S3 C?2 <4 



+ 2 6^1 c?i c?2 ^3 + 2 -^2 ^2 ^3 ^4 H~ 2 *3 d^ d^d^-\-2 .S4 <;?4 ^5 c?, + 2 .S5 dr^ d^ «?2 



+ 2 5, <?3 ^4 ^5 + 2 .S2 C?4 C?5 C?i + 2 .S3 <4 ^j «;?2 + ^ ^4 C?, C?2 ^3 + 2 ^5 <i?2 ^3 ^4 



+ 2 <f 1 c?2 ^3 <^4 + 2 fi?2 d^ d^d^-\-2 d^ d^ d^di-\-2 d^ d^ J, <?2 + 2 d^ d^ ^2 d~^ ; 



>. (130.) 



