...{D.\EABC} = {D'.E'A'B'C1}, [E.ABCD] = {E' .A B' C' D'} must in all cases exist together; which, when A and A', B and B', C and C", D and D' are interchangeable, •would be manifestly impossible unless also E and E' were interchangeable ; and...

...when consequently the several segments intercepted on L by the several pairs of corresponding rays A and A', B and B', C and C', D and D', &c. of the two pencils, have a common magnitude or middle point) differ in no respect from the general...

...and e', c' and_/" the three pairs of opposite chords intercepted upon them by the given quadric ; and A and A', B and B', C and C', D and D', K and E', ¥ and i" the six pairs of points which divide harmonically the six pairs of segments a and...

...particular circles of the system which arc touched by the same circle 0. 7. If four pairs of points A and A', B and B', C and C', D and D', un any right line be such, that any two corresponding constituents mav be interchanged without violating...

...consideration at a and c on the perimeter of A, and b and d on that of B. Cut at these points and call the ends a and a', b and b', c and c', d and d' ; a,b,c and d remain fixed. Now revolve through 180° about the axis AB, and join the free ends. Before...

.../-conies. Starting as above with the correspondence (a, b, c, d), (a', b', c', d"), if the intersections of a and a, b and b', c and c', d and d' are called A, B, C, D respectively, then we have A, B, (7, D for a tetrad of /-points, lying on the /-conic...

...four pairs of adjacent angles. Other pairs of angles have also received special names. Thus the angles a and a', b and b', c and c', d and d' are called corresponding angles. The angles a and d', b and c', c and b', d and a', are called alternate...

...corresponding side of the other. Let the equiangular polygons ABCD, A'B'C'D' have the corresponding angles A and A', B and B', C and C', D and D', occurring in direct order. Let the sides AB and A'B' be parallel. To prove that BC is parallel to B'C',...

...proportional. D> C' B' A* HYPOTHESIS. The parallels AD and A'D' are cut by the transversals OA, OB, OC, OD, at A and A', B and B', C and C', D and D'. CONCLUSION. A'B' B'C' C'D' PROOF Since A'D' is II to AD, the A OAB and OA'B', OBC and OB'C', OCD and...

...identical with that present in the original fertilized egg; and, as in the egg, they form four pairs, A and a, B and b, C and c, D and d, each consisting of a maternal and a paternal member. This mode of development continues throughout...