for the Development of the Function <3 (a + x.) 215 



From the two equations 



(1). Sin, a + X. Cos. x — Cos. a + x. Sin. x = Sin. a. 

 (2) . Cos. a + X. Cos. X + Sin. a + x. Sin. x = Cos. a. 



we can easily deduce two other equations which have 

 been usually considered as the foundation upon which the 

 whole theory of these functions rests. These equations 

 are 



Sin. a + X = Sin. a. Cos. x + Cos. a. Sin. x. 

 Cos. a + X = Cos. a. Cos. x. — Sin. a. Sin. x; 



and to have a demonstration of them, we multiply equation 

 (1) by the cos. x, and equation (2) by the sin. x: then by 

 adduig these two products together, we shall find Sin. a + x. 

 (Cos^. X + Sin*, x) = Sin. a. Cos. x + Cos. a. Sin. x. 

 and as the sum of the squares of the sine and cosine is 

 equal to the square of the radius (equal to unity), we have 

 Sin. a + X = Sin. a. Cos. x + Cos. a. Sin. x. Again, if 

 we multiply equation ( 1 ) by the sin . x, and equation (2) by 

 the COS. X, and if we subtract the latter product from the 

 former, v/e shall have Cos. a + x. (Cos^ x + Sin^. x.) = 

 Cos. a. Cos. X — Sin. a. Sin x; or Cos, a + x = Cos. a. 

 Cos. X — Sin. a. Sin. x. 



The examples which we have now given evidently show 

 the advantage that may sometimes be derived from the de- 

 velopment which we have here investigated; and it would 

 not be difficult to adduce other instances to which it might 

 be successfully applied. But as we have accomplished what 

 we originally proposed; namely, to point out the analytical 

 distinction betwixt the two theorems of Bernoulli and Taylor, 

 we shall leave the further consideration of the subject to 

 those who may think it worth the trouble of a more minute 

 examination. 



W. S. 



4 xxxin. On 



