278 Prof. J. N. Lockyer. [June 21, 



The general result is that there may be as many forms of solution 

 as there are variables (the differential equations being of the first 

 order, to which they may always be reduced). Each form is derived 

 from the one before by the process of finding the envelope, and each 

 contains fewer arbitrary constants by one than the form from which 

 it is directly derived. 



The general theory is given in 2 for the case when the differential 

 coefficients are given explicitly in terms of the variables. In 3 it 

 is extended to the case when they are given implicitly, and in 4 it 

 is shown how the singular solutions are to be formed from the differen- 

 tial equations themselves. In 5 9 the theory is connected with 

 that of consecutive solutions belonging to the complete primitive. 

 10 13 are taken up with geometrical interpretations relating to 

 plane curves, and also to curves in space of n + 1 dimensions, n + 1 

 being the number of variables. In 14 16 the case is discussed in 

 which a system of singular solutions is included in a former system or 

 in the complete primitive. 



The rest of the paper contains the application of the theory to 

 certain examples. The first example ( 17 21) is the case of the 

 " lines in two osculating planes " of a twisted curve, and in particular 

 of a twisted cubic. The particular example is given by Mayer and 

 Serret. The second ( 22 26) is that of the congruency of common 

 tangents to two quadric surfaces, and generally ( 27 38) of the 

 bitangents to any surface. The third ( 39 49)is that of the essen- 

 tially different kind of congruency which consists of the inflexional 

 tangents to a surface. It seems natural to call these two kinds of 

 congruency bitangential and inflexional respectively. The fourth ex- 

 ample ( 50 52) is that of a system of conies touching six planes. 

 The fifth ( 53 60) is that of a doubly infinite system of parabolas 

 in one plane, the differential equation being a case of an extension of 

 Clairaut's form y = px+f(p), which is explained in 53 55. 



XII. " The Spectrum Changes in ft Lyrse. Preliminary Note." 

 By J. NORMAN LOCKYER, C.B., F.R.S. Received June 13, 

 1894. 



The spectrum of this well known variable star was first investi- 

 gated photographically by Professor Pickering, at Harvard College 

 Observatory, and a preliminary account of the results was published 

 in 1891.* Dark and bright lines were found to be associated in the 

 spectrum, and further, the bright lines were found to change their 

 positions with respect to the corresponding dark ones according to 

 the interval of time which had elapsed since the preceding minimum. 

 * 'Ast. Nach.,' 2707; 'Observatory,' 1891, p. 341. 



