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of the generatrices has begun in the plane ZOV-^ on the line 

 — the maximum pitch >j has been measured out; on the foHowing 

 lines I — 1, // — 2, etc. the succeedino' pitches are now continually 

 measured out from the nodal line, the measuring ceasing in the 

 position where the lotation amounts to 180°, thus the whole 

 height of the cylindroid having been described on both sides of the 

 centre 0. To explain this we make use in the first place of the 

 circle, wliich has served to find the length of the pitch of any 

 generatix, drawn on half the scale of the principal figure, where the 

 length of any generatrix (e. g. II— 2) is indicated with its corres- 

 ponding angle of deviation. 



In the second place we sec axonometrically constructed tlie pitch 

 curve projected on XOY with the projections O^O, 0—1 etc. of 

 the generatrices. Further is drawn the perpendicular ^1 S to 11— 4 

 passing through A on a^l — 7. According to 2b this is a screw, 

 so it meets tiie line «'^7—5 conjugate to 1 — 7. 



Remarks, a. The projection of the above mentioned pitch curve 

 lies entirely on one side of the axis Y. Evidently this is lialf the 

 figure we obtain in constructing the curve with tlie equation : 

 n ^z a -^ 2r COS' 0. It is merely a consequence of our peculiar manner 

 of measuring that by the followed construction only half the figure 

 is obtained. 



h. In the constructed figure all the pitch values have the same 

 sign : if tliis were not the case, the figure of the projection would 

 be changed, the curve half drawn, half Jotted wouhi show a double 

 point, so the entire projection a fourfold point. This last has now 

 become isolated. 



•j. Fig. 2 represents the parallel projection of the principal curve 

 of the representation of CaPOUAli and must be considered in con- 

 nection with fig. 1. On the generatrix a of C^ a point A has been 

 fixed. Through -1 the screw A A' B has been diawn, being one of 

 the rays of the pencil through the centre -4 in the plane A a' ^ a. 

 J, as pole of «, determines with (c a linear system of the iJ'i order 

 of linear complexes. 



With these figuies we suppose that C^ lies in the space .^, the 

 principal curve in the conjugate space JS", . To any linear complex 

 in .^ a plane in -S^j corresponds, to any screw i point. The prin- 

 cipal curve is the locus of the points in J^i to which corresponds 

 in .y not only a single screw but a pencil of screws. Ac- 

 coidinn' to these conventions and to the indication sub N'\ 1 



