traced upon the Surface of the Sphere. 307 



certain assignable portion of a sphere, minus an assignable square, this me- 

 thod not giving that result, does not apply to the case in question. 







The same remark may be made respecting his third solution, or that 

 which is expressed by the general form of equation (2.). If we take - 



772. 



equal to the requisite spherical lune, then sin <p will be irrational and 



transcendental. So that, in this case neither the curvilinear area nor the 

 residue of the spherical surface will fulfil the condition. And if we take 



</> TT, 2 TT &c., we still have TT, - . 2 T, which will not fulfil the 



n n 



conditions neither. We see, then, that this method also fails, except in 

 particular relations amongst the constants m and n. 



Q 



3. When -j-, we have the spiral of PAPPUS, that is the curve by 



T? 



which he abstracted such a portion of a sphere as was perfectly quadrable. 



Here 



d A = (1 cos (j)) 4 d (p, or 



A = 40 4 sin -4" const ...................... (4>.) 



rjr 



Now, when = -. we have A = 2 TT 4, 



which, subtracted from 2 ir, leaves, as before, for a residue of the hemisphere, 

 the square of the diameter of the sphere. This agrees with the usual solu- 

 tion of the problem. 



a 



4. Let (f> = : then A = 2 <p 2 sin </> + C. This taken between 



S8 



and TT gives A 2 tr, or the spiral bisects the superficies of the sphere. 



5. Take <p i $ : then A S <p 8 sin ; and this taken from to 



TT 



to - j- gives 4 TT 8 ; or the residue of the whole sphere equal to eight 



A 



times the square of radius. In like manner, several of the values of - may 



be found, which, when </> is taken between certain limits, the residue of the 

 sphere, or of particular portions of it, shall be quadrable. 



