498 PROCEEDINGS OF THE AMERICAN ACADEMY 



Substituting this value of c in [4], we obtain 



D (a x ) = log, a-a x ■ Dx. [e] 



The Differentials of the Trigonometrical Functions. 



Let z = x-\-h, then Dz= Dx, [1] 



also sin z = sin x cos h -\- cos x sin h, 



= cos h ' sin x -\- sin /j y/ 1 — sin 2 cc. 



n/ . . , rw • \ i •? — 2 sin x D (sin z) 



x> (sin z) = cos A • D (sin a:) -J- sin n . — • 



■* V 1 — sin 3 x 



= [cos h — sin h • \D (sin x). [2] 



COd JO I 



Combining [1] and [2] 



D (sin z) D (sin #) cos A cos x — sin h sin x 



D z D x cos x 



cos 2 Z> (sin x) 

 cos a: D x ' 



or, separating variables, 



[3] 



1 ■ D (sin 2) 1 ^ J> (sin z) 



cos z D z ' cos a; Dx ' 



D (sin x) = c ' cos x • D x. [4] 



To determine c (x being the circular measure of the angle) 



Put cos x = sin I -- — x j. 



By [4] D (cos x) = c • cos ( ^ — a: ) D (^ — xj , 



or D (cos a;) = — c sin x D x. [5] 



From [4] and [5] D (tan a;) = c sec 2 x D x. * [6] 



Now in [4] c cannot be greater than unity, for if it were, D (sin x) 

 would exceed D x for all values of x less than a certain value. 

 Hence, x and sin x starting together from zero, sin x would, for these 

 values, exceed x, which is impossible. 



Again, from [6] c cannot be less than unity, for then D (tan x) 



