XU.11>> ON M Ul"A< I>. 341 



The equations marked (5) occur for the firsl time in La- 

 grange's memoir of 177 J referred to in the historical sketch 

 which precedes this essay, and arise not by any derivation 

 from their mechanical meaning, I nit simply as analytical 

 abridgments naturally presenting themselves in the course of 

 bis investigations, and then afterwards examined and defined. 

 The values of these velocities dP. dQ. till, may however be 

 obtained from their definitions without calculation, by means 

 of the following simple consideration, — that the velocity round 

 any one of the axes is the same with the velocity of a point 

 (distant unity from in a ueond axis) estimated in the direc- 

 tion of the third axis. Thus the components, in the direction 

 of the fixed axes, of the velocity of the point (a. a . a ) in the 

 axis of x t , being da, da. da , its velocity in the direction of 

 the axis of y will he bda -)- b da -\- b da , which is therefore 

 equal to dli. the velocity of rotation round the axis of z,. The 

 velocities dP, dQ are then had by changing the letters. 



The following corollaries from the above formulas will be 

 useful on a variety of occasions: (7) 



du> h- da '-h da'" = dQ'+dR* 

 db +. db ■ 4- db ■ = dW + d/> 

 dc' + def+da" = dP'-^dQ' 



dadb — d,i db + da db = — dPdQ 

 dbdc + db de + db de = — dQdR 



deda -f- de da -+- (/<■ da = — dlidP 



ad a -4- a d<i -+- a d'a = — (dQ" -+- dR>) 

 bd b -4- 6 db -+- b d b" ss — (dli -h dP>) 

 rd C — C (I C -+- C de = — \(IP' ■+■ dQ*) 



ad h + „ db -j- a ,/ & = efiPdQ — d It 

 bd c + A rfc -t- A rf*£ = dQdR — d P 

 rd a __ c tfa ■+■ c «fa = dfidP — efQ 



ad c _|_ a de + a cfr = dK«*P + ef Q 



&<p a .+. 6 <f a h_ 6 d-a = <//'dQ~ (IH 

 c d*b -+- e d*b l -4- c db = dQdli 4- <f P 



