the Differential and Integral Calculus. 27 



due attention to the matter, is the object of the preceding and 

 subsequent remarks. 



Up to a recent period the fluxional notation was commonly 

 used by English mathematicians. Mr. Woodhouse assigned 

 reasons for the adoption of the differential instead of the flux- 

 ional notation in the preface to his Principles of Analytical 

 Calculation, published in 1803; he employed the differential 

 method in an elaborate paper published in the Philosophical 

 Transactions in the next year : previously he had used the 

 fluxional notation. The English translation of Lacroix was 

 published in 1816; the differential notation first occurred in 

 the Cambridge Problems in 1817. I believe its first appear- 

 ance in any English mathematical periodical was in the second 

 volume of the Mathematical Repository, in a solution by Mr. 

 Ivory. 



In the translation of Lacroix's Differential and Integral 



Calculus just named, it was laid down that if u be a function 



d u 

 of x and u = ax 3 , then d u = 3 a? d x, and -=— = 3a^ 2 ; the 



first expression was termed the " differential" of the equation, 

 the latter was called its " differential coefficient." 



I believe this notation has generally been since used by 

 writers on the differential calculus, both in England and else- 

 where ; another mode of differentiating, however, has been par- 

 tially adopted at Cambridge, or perhaps it may be more ac- 

 curately termed a substitute for differentiating ; it has been 

 called " the calculus of differential coefficients:" instead of 



writing -^ — , for the differential coefficient as above, they 

 ° dx 



write d x u : if u and k be functions of a-, they write 

 dj, (uz) = u dj. z + z dm u ' 



d u 

 Similarly, d x (u z ) = u z (z — f- log, u d x z). 



The radius of curvature is thus expressed : — 



'• = --s^{ 1+ (^) 2 } f - 



The equations of motion are thus written : — 

 d t x = velocity parallel to x. 

 d t y = velocity parallel to^. 

 d t z = velocity parallel to z. 

 d?x= X d?y =Yd?z= Z. 

 It is well known that these expressions are usually written, 

 d x d 2 x - 



Perhaps Mr. Jarrett's paper on algebraic notation in the third 



