SECT. VI.] GEOMETRICAL ILLUSTRATION. 197 



and should differ from it only by a quantity which becomes less 

 than any given magnitude: this limit is the value of the series. 

 Now we may prove rigorously that the series in question satisfy 

 the last condition. 



229. Take the preceding equation (X) in which we can give 

 to x any value whatever; we shall consider this quantity as a 

 new ordinate, which gives rise to the following construction. 



Having traced on the plane of x and y (see fig. 8) a rectangle 

 whose base OTT is equal to the semi-circumference, and whose 

 height is ?r ; on the middle point m of the side parallel to the 

 base, let us raise perpendicularly to the plane of the rectangle 

 a line equal to |TT, and from the upper end of this line draw 

 straight lines to the four corners of the rectangle. Thus will be 

 formed a quadrangular pyramid. If we now measure from the 

 point on the shorter side of the rectangle, any line equal to a, 

 and through the end of this line draw a plane parallel to the base 

 OTT, and perpendicular to the plane of the rectangle, the section 

 common to this plane and to the solid will be the trapezium whose 

 height is equal to a. The variable ordinate of the contour of 

 this trapezium is equal, as we have just seen, to 



^ sm 3a sm % x + 7z sm ^ a sm 



O O 



(sin a sin x 



7T \ 



It follows from this that calling x, y, z the co-ordinates of any 

 point whatever of the upper surface of the quadrangular pyramid 

 which we have formed, we have for the equation of the surface 

 of the polyhedron, between the limits 



1 sin x sin y sin 3x sin 3^ sin 5x sin oy 



-TTZ = -- j2 32 - ^2 - ^- 



