552 MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES. 



or negative ; and also that on the ordinate which passes through the intersection 

 of the one pair of lines, the distance intercepted between the other pair is the 

 linear unit. Also, it may be proved that for the ordinates at (n + ^)II the inter- 

 cepted distances are alike, each being represented by V^ . 



50. Leaving, for the present, the consideration of the four fundamental func- 

 tions, we may give our attention to the differences 1 — 1 and 1 — 1, 

 which we have already agreed to represent by the symbols cos t and sin t . By 

 subtracting the value of [t + u) from that of {t + u), as given in article 45, 

 we find 



0(« + «)-0(« + «) = (0*- H«H0«-[l]«} - flll« - S*JfQ« - m«i 



that is 



cos (t + u) — cos t . cos u — sin t . sin u , 



and similarly we can obtain 



sin (t + u) — sin t . cos u + cos t . sin u , 



so that the computation of the values of these compound functions may be made 

 independently of those of the elementary functions, almost as we have already 

 done for the binary functions sus t and cat t . 



If, having laid off OA (fig. 12), equal to the linear unit, we make OC 



equal to the difference £ — £, and set up the perpendicular CD equal to 

 * — * ; then we have OC 2 + CD 2 = OA 2 , wherefore the point D must be in the 



