SOLIDS OH 81 RF \< E8. (7 , 



D'Alembert's method of integrating simultaneous lin- 

 ear equations is regarded by sonic of the first mathematicians 

 of Europe as the best, and I have therefore introduced the 

 equations (:35) ; but if the direct substitution of exponential 

 functions of the time be preferred (a method which has often 

 the advantage of greater expedition), it would ool be access- 

 ary, to form these equations, as the values of d \ . d r ,. d £, are 

 derivable from their equations of definition (10) in terms of 

 (lie rotatory velocities and the coordinates, parallel to the 

 body-axes, of the centre of gravity. For if we multiply h\ 

 n. a . it . thi' values of the second differentials of £, £',£", the 

 -urn of the three products will be equal to d i by the defini- 

 tion of this quantity, which is in fact the velocity which the 

 point O gains in every interval d/ estimated in the direction 

 which the hoily's first axis has at the beginning of thai inter- 

 val. It is because this acceleration is measured not on the 

 variable axis itself, but on the direction which that axis had at 

 the beginning of At. that the sum of the elements d% will 

 not make up the velocity d£„ nor the sum of the elements ilE 

 the finite rate of increase of £,. In consequence of these dis- 

 tinctions, many difficulties might arise in considering geome- 

 trically problems of the nature of the one before u> : hut tiny 

 in always either avoided or explained by the adoption of 

 analytical methods of solution, and I feel assured that the ex- 

 perience of those who are conversant with these methods will 

 bear me out in saying that the necessity of even adverting to 

 the difficulties of geometrical mechanics disappears precisely 

 in proportion to the purity and generality of the analysis. 

 While on this subject however I ought to remark that in 

 consequence of this incompleteness of the values n \' d{- . >lr . <U . 

 and in the case of perfect rolling of rf£, il^ . <l- . the applica- 

 tion of Lagrange's Subsidiary Formula (Mic. ./mil. Vol. I. 



p. 313) is inadmissible in Mich cases, and would lead to fals. 

 results oven if the velocities </-'. d%\ <h be expressed in func- 

 tions of the finite angles i. $, 8 and their velocities. In short 

 his method is applicable only when the differential equations 

 connecting the variables fulfil the conditions of integrability. 



