f>18 MR EDWARD SANG ON THE THIRD CO-ORDINATE BRANCH 



to give significance to the symbol T y# ; there may be any number of terms inter- 

 polated into the above series, and we cannot say that f is not one of the endless 

 numbers of terms between sin t, cos t, — sin t, — cos t. 



In a paper which I gave in the Annals of Philosophy, for August 1829, it is 

 shown that if u and v be two functions of some primary, which for the sake of 

 conciseness we may suppress, the n th derivative of their product uv, takes the form 



n (uv)= n u . v + j n _ x u ■ 1 v + l -y- n _ 2 w . 2 v + etc. 



which is quite analogous to the n th power of the binome u + v; and it is also shown, 

 that the n th primitive (or integral) of the same product is 



. N n n n + 1 



- n (uv) = _ n u.v- j _ {n+1) u 1 v + i -^- _ (n + 1) u. 2 v- etc. 



which is also the counterpart of (u + v)~ n . 



It is thus seen that these expressions are true whether a positive or a nega- 

 tive value be assigned to n, and the question may well be propounded, " Do these 

 formulae hold good when n has a fractional value V 



Without entering farther into this subject, it is enough for me to repeat the 

 remark, that in the present state of our knowledge, we can form no idea of such 

 fractional derivation; and that, therefore, the fourth branch of the general 

 problem, viz., that in which the order of derivation is sought for, has for us 

 scarcely any significance. There remain, then, only the three branches, the 

 calculus of differentials or derivatives, that of integrals or primitives, and lastly 

 the calculus of primaries. 



The great majority of problems in theoretical mechanics belong to this third 

 branch. The velocity of a moving point is generally dependent on its position; 

 now, the velocity is the first derivative of the position regarded as a function of 

 the time, and thus the problem really is, " from the known relation between 

 the function and its derivative, to determine the primary, viz., the time." This 

 problem is commonly resolved by a very simple process, — the first derivative of 

 x, regarded as a function of t, is the reciprocal of the first derivative of t, regarded 

 as a function of x ; and so, by causing t to appear as the function, and x as the 

 primary, we convert the problem into another belonging to the integral calculus. 

 This convertibility of the problem has somewhat concealed its true nature. 



The first example which we have on record of the use of the laws of change in 

 scientific research, belongs to this third branch of the subject, and clearly exhibits 

 its true nature. Nepair regarded his two flowing quantities, his Artijicialis and 

 Naturalist as connected by this law that, while the velocity of change in the 

 former is uniform, that in the other is variable, and proportional to the Naturalis 

 itself. In modern language he made the Artijicialis his primary variable, and. 

 prescribed the condition that the first derivative of the function should be pro- 



