274 G. H. KNIBBS. 



real for plus values of x, and two imaginary for minus values. If 

 we take z = iy perpendicular to the xy surface, 1 the several branches 

 can be represented as continuous on what may be called a double 

 Cassinian lore, or lemniscate double-tore, that is to say, two solid 

 rings in contact and of such form that the section is a lemniscate. 

 The polar equation of this section being p 2 = cos 29, gives two 

 tangents perpendicular to one another, the axes y and z. Figures 

 4 and 5 shew the development of the surface. The plane of the 

 lemniscate 00"0'" contains the axes YY and ZZ, and the axis XX 

 is perpendicular thereto. The positive branches lie in the surface 

 xy and the negative in the surface xz 2 The double toroidal 

 surface will allow the previous curve, (13), to be also continuously 

 represented, but its various branches will unite somewhat differently 

 in order to include the imaginary parts of the curve; contrast Figs. 

 3 and 6. In the latter figure the curve, Z' O etc., lies in the xz 

 plane, as also does the curve from A to B. From to A, and from 

 B onward, the curve is in the xy plane. It will be noticed that 

 the double toroidal surface requires the continuity to be established 

 in the order shewn by the dots; Fig. 6. 



By making the radius of the sphere, or the parameters of the 

 curves, of a higher order of infinity, the continuity is developed to 

 a higher order of infinitesimal, and to this process there is no con- 

 ceptual limit, that is the pure plane is of absolute-zero-curvature, 

 or rather, has no curvature at all. 



14. Infinitesimal approximation and absolute identity in differ- 

 ential coefficients. — We have seen that the resolution of a discon- 

 tinuity through infinite paths gives nothing more than pseudo- 

 continuity, and that conceptually we can always recognise the 

 purely formal and artificial nature of the continuity thus apparently 

 attained. In practical applications the matter is of no moment, 



1 See § 33 hereinafter. 



8 In Fig. 5 one tore lies or fits within the other. In order that the 

 infinities should be identical in magnitude, the loop of the lemniscate 

 ought to be equal in length to the circle OXO'X, hence one must be above 

 the other as in the shaded part of Fig. 5. It is easily shewn that the 

 length of the loop is expressed by an elliptic integral of the first kind. 



