454 EEPORT — 1882. 



d d 

 whence p -— sin a= 'v/(p^ — sin*«). 

 a f} 



The integral of this equation defines the involute 



6 = cosec a^y{p'^ — sin- a) — cos ( — sin a \ 



After replacing the values of p, 6, this becomes 



tan-' ( — L ) = cosec a^icoa"^ r + sin- p - cos- a) — cos"^ V^ ^ 



\cosrJ ' (cos^ r + sin'^p). 



sin a 



6. If sin « = \, or the circumference of the rolling circle is douhle that of the fixed 

 circle, the epicycloid is the spherical cubic with a triple focus and triple cyclic arc. 

 tan- y (tan .r - ^/3) + (tan x -\sj^y = 0. 



6. The involute of a small circle is rectitiable. 



c,. sin p . d 6 J • o • o .9 



bince — — - •= sm ?• — ^ and sni- r — sm- p = sm^ a. 

 sin »• as , 



ds sin a = sin ?• dr, 



8 sin rt=cos a — cos r. 



The arc varies as the distance of the moving point from the plane of the small 



circle. 



7. On the quadratui-e of areas enclosed by the involute of a small circle. 



Two spaces are selected, both bounded by the fixed circle and its involute ; but 

 the tliird limit may be (1) the rolling great circle, or (2) the radius-vector of the 

 moving point. 



In (1) an elemental area has been chosen by Olairant, which is comprised 

 between the involute and two consecutive positions of the generating great circle. 



If 6 be the arc subtended at the fixed centre by the arc which has been 

 traversed, and (f) the equal length of the generating great circle, then 



6 sin <i = (j), cos r = cos a cos (f). 



The area in (1) = c? ^ (cos a -cos ?•) =cot o; \d (f> (1 -cos <^) =cot a (0 — sin (f>). 



Clairant remarks that if from this area a spherical sector be subtracted with 

 (j) for its vertical angle, the remainder would he strictly quadrable. 



In (2) the area = cosec « d?- (cos a — cos r) cosec" r sin* p 



= cot a ((^ — sin (^ ) — cos a tan -■ (tan <f> cosec a) +tan -' (sin <^ cot «), 

 since cos r = cos a cos (j), sin ^j = cos a sin (^. 



From this area also, if a certain spherical sector be subtracted, the remainder is 

 strictly quadrable. 



8. Bernouilli remarks that the apparent path of the sun's centre is an epitro- 

 choid, affiliated to the rectifiable epicycloid or involute of a small circle. 



' If the sun were to move in the ecliptic with a velocity equal to that which 

 the tropic of Capricorn has to execute the diurnal motion, and so the time of a 

 revolution of the sun in the ecliptic were to the time of revolution of the sphere, 

 which constitutes the length of the natural day, in the same ratio which the radius 

 of the ecliptic (or of the sphere) bears to the radius of the tropic as 1 : cos 23°30', 

 the centre of the sun would exactly describe a rectifiable epicycloid. But as the 

 sun has its proper motion in the ecliptic much slower, the line which the centre of 

 the sun describes between the two tropics during the space of a year by the com- 

 bined common and proper motion will he a cydoide allangee, rather than a spiral, 

 under the form that Tycho conceived.' 



9. The preceding paragraphs are extracted in a condensed form from a memoir 

 on spherical, cycloidal, and trochoidal curves, which will appear in the forthcoming 

 number for October 1882 of the ' Quarterly Journal of Mathematics.' 



8. On a Fartial Differential Equation. By J.W. L. Glaisher, M.A., F.B.S. 

 In the British Association Eeport for 1878, p. 469,' it is shown that 

 « = e'iV(..'-+. A) ig a particular integral of the partial differential equation 



• See also PMl, Trans. 1881, p. 774. 



