196 



Sir William Thomson. 



[Nov. 26, 



3. Let us now simplify by choosing our coordinates so that the 

 values of 0, %, &c., are each zero for every position of the path 

 P' P ; and let ^, for any position of this path, be the action along it 

 reckoned from zero at P'. These assumptions, expressed in symbols, 

 are as follows : 



j u/(p u/x r r " 



(jfor all values of y- and V'', i{ = >X = > ! 0'=0, x' = 0, J 



..(3). 



4. Taking now 



i/r = o, y^ = oY'} o0 = o, ox ' > <>Y r ' =: o^' = > ox' > > 



..(4); 

 we have 



A( y-, o0, o% - * oY^'j o0', ox' ....) A ( y, o, o, . . . . , o, o, o, . . . .) (5) 



and, in virtue of this and of (3) and (1), (2) becomes 



A( ^, 0, X, , 0, 0', x' ....)= A( y-, 0, 0, . . . . , 0, 0, 0)" 



i' 2 +55 X ' 2 +663' 2 



(6), 



' +3633' 



where, merely for simplicity of notation, we suppose the total number 

 of freedoms of the system, that is to say, the total number of the co- 

 ordinates yr, 0, x, 3, to be four ; and for brevity put 



(m = U, (,-) = !*, (^=22,40. 



\^0V \d^dx/ oVx 



(7). 



5. From (6) we find, by (1), 



f = 110 + 12 X + 133 + 140' + 15x' 



= 310 + 32 X + 333 + 340' + 35 X ' + 363' 



rf = 510 + 52 X + 533 + 540' + 55 X ' + 563' 

 T = 610 + 62 X + 633 + 640' + 65 X ' + 663' 



(8). 



