MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES. 553 



circumference of a circle described from as a centre with OA as a radius. Pro- 

 ceeding now as in article 19, let us suppose the point D to be carried to a small 

 distance along the curve, then the increment of the sector AOD must be repre- 

 sented by I {OC . SCT> - CD . flOC} , that is by £ {(0 t - tf + (0 2 - tf} St 

 or by ^St, and consequently the double of the area AOD must represent the 

 primary variable 2. Also, since the increment of the curve is, in all cases, 

 -v/{<5CD 2 + SOC 2 }, and since SCT> = OC . St, 80C = — CD . 8t, it follows that 

 the increment of the curve is just St, so that the arc AD stands for the primary 2. 



Now when t is made equal to the above value of II, the function 1 — 1 

 becomes zero, so that the point C is then at ; wherefore II must represent the 

 length of the quadrantal arc AB. The number tt is therefore half of the well- 

 known value 7T = 3-14159 26536 for the length of the semicircumference of 

 a circle of which the radius is unit ; and therefore, moreover, the functions 

 1 — 1 and 02 — 02, are in reality the cosine and sine of the arc 2 . 



In this way we may imagine the whole doctrines of trigonometry as imported 

 into our investigation, and as forming a scholium to the fourth case of the first 

 problem in the calculus of primaries. 



51. Having now arrived, in the regular course of our inquiry, at the measure 

 of angular position, and at the circular functions, we may resume the considera- 

 tion of the angular motion of the trigon ABC (fig. 4), which was left off in 

 article 42. 



If we denote the inclination of the base BC by 6, we have 



sin 



= -an = g^ 3 • e 2 j A 2 — A * | whence 



COS 



. z, 2 3A* 2 - 6 A* • At + 3A* 2 



~ 4 { A* 2 + A* 2 + A? - At - At- A- At- At- At] 



_ 4 A^ 2 - 4A^ • A^ - 4A^ • A* + At 2 + 2 At • At + A* 2 



4 A* 2 - 4A* • A* - 4A* • At + 4: At 2 - 4A* . A* + 4A* 2 



and consequently 



cos0 = ^e [2 A* - A* - A*} 



Taking the differential of sin 0, we have 



e .se = * j$.e ¥ (2a$- a* - At] St 



= 2 a/3 . cos 6 . St wherefore 



S6 = * J3 . St and 6 = t *f- ; 



cos 



