On Machines in General. 3 1 



of a triangle, trigonometry may furnish the means of 

 giving a great number of different forms to the fundamental 

 equations (E) and (F) ; and I shall content myself with 

 indicating one of them, which is remarkable on account of 

 the method contrived by geometricians, of referring move- 

 ments to three plans perpendicular to each other; which 

 gives a great deal of elegance and simplicity to the solutions. 

 Let us imagine, therefore, at pleasure, three axes perpen- 

 dicular to each other ; and let us conceive that the veloci- 

 ties W, V, U and u, are each of them decomposed into 

 three others parallel to these axes. This being done, let 

 us call 



Those which answer to W, 



Those which answer to V, 



Those which answer to U, 



Tliose which answer to u, 



Now if we pay a little atteniion, we shall easily see that the 

 first fundamental equation (E) mav be placed under this form, 

 smVU' + smV U" + .>' m V" U'" = 0; and the second 

 (F) under the latter s vi u U' + s ?w ti' U" + 5 ra li" U"' = o ; 

 because in general every quantity which is the product of 

 two velocities A and B, by the cosines of the angle com- 

 prehended between them, is equal to the sum of three other 

 products A' B' + A' B ' + A'" B" ; A' A" A'", being the 

 estiu ated velocity A of these three axes, and B' B" B"' being: 

 the estimated velocity B in the ratio of these same axes : 

 i. e. A' being the velocity A, and B' the velocity B, esti- 

 mated parallel to the first of these axes : A" and B" the same 

 velocities A and B' estimated parallel to the second axis, 

 A"' and B'' the same velocities estimated parallel to the third 

 axis : this is easily proved by the elements of geometry. 



In the case of equilibrium, the first of these transformed 

 ^c^uations is reduced to = 0; and the second, because in 

 this case W = U, becomes s m u' W + s mu" VV + s m 

 u'" "W" = 0; which expresses all the conditions of the 

 equilibrium. 



Wtiea the movement changes by insensible decrees, we 

 have found (XXV.) that the fundamental equations become 

 s mW p t cosine B. — smV dV =0, and smupd I cosine 



T — smud 



