100 CELESTIAL MECHANICS. I. i. 1. 



had been required to express the relation of 5y to ^x 

 while y remained evanescent, the proposition would have 

 failed, since in this case 5jr is initially =0, and varies at 

 first as the square of 3y, the first term of the Taylorian 

 theorem, on which tlie reasoning is founded, here vanish- 

 ing altogether : but when the application is clearly under- 

 stood, the argument is readily admitted almost as an 

 axiom. 



Case 2. When the forces concerned are not at right 

 angles to each other, they may both be referred to ortho- 

 gonal coordinates, and if their projections in any three 

 such directions be a, b, and c, these lines will represent 

 the respective portions of the force V (a^ -\-b^ +c^): and 

 if the second force be represented by similar ordinates 

 a\ b\ and c\ the forces may be combined by adding toge- 

 ther their constituent portions, as reduced to the same 

 directions, giving together a + a\ b-\- b\ and c + c', which 

 may again be combined into a single force ; and this force 

 z will always be represented by the diagonal of the paral- 

 lelogram, formed by lines representing the two former, x 



and y: and in the same manner 



y^^-^^^^i I, ' ^'^y greater number of forces 



^.f:::::^!^^^^..-.... \ may be combined, by reducing 



^'yjy^ \ 5 them to three orthogonal direc- 



,^C^,„jx!, \ tions, and by adding together 



^ ^ their respective results. 



250. Theorem. When several forces act 

 on the same moving point, if we suppose the 

 place of the point to be changed in any 

 manner whatever to a minute distance, the 

 product of the joint force into this distance 



