496 Proceedings of the Royal Society of Edinburgh. [Sess. 
on the y>-discrirninant of a differential equation of the first order, Chrystal 
begins by pointing out that Cayley, in his early discussions of singular 
solutions, was led to a proposition which was erroneous, or at least very 
misleading. Cayley stated that when the differential equation had no 
singular solution it did not admit of an algebraic primitive. Chrystal, on 
the other hand, showed that although it is true that, when a singular 
solution exists, the primitive is algebraic, yet it is the exception and not 
the rule that there is a singular solution when the primitive is algebraic. 
It is interesting to note that there has been a steady demand for copies of 
this particular paper on the jj-discriminant and the connected theory 
of envelopes. 
The outstanding characteristic of these mathematical papers is the 
endeavour after rigorous demonstration. They are clearly written, and 
are put together in a way which shows a keen perception of the essential 
nature of the problem contemplated and the inborn power of the teacher in 
presenting his demonstrations so as to be easily followed by the intelligent 
student. Chrystal, indeed, possessed the intuitive art of the born teacher. 
Wide and deepty read in all that was best in mathematical literature, 
possessing at the same time an artistic appreciation of the beauty of form 
and logical sequence, he never failed to impart to his presentation of a 
mathematical argument the distinctive personal touch which appeals to the 
real student. The following sentences drafted by his eldest son, Mr George 
Chrystal of the Home Office, give what may be regarded as Chrystal’s 
own estimate of himself in the ranks of mathematicians : — 
“ My father in familiar conversation with me always declined the title 
of a great original mathematician. How far this was justified, I have no 
means of judging ; but his real bent seemed to be towards physical science 
— towards the concrete rather than the abstract. With this, however, he 
had a keen appreciation, a great knowledge, and a thorough understanding 
of what had been achieved by the giants of the mathematical world — the 
Cayleys and the Riemanns, whose results, as he used to tell me, were some- 
times reached by stages and processes which even these great men themselves 
could not always thoroughly explain or account for. 
“ What he regarded (I believe, though he never told me so in terms) as 
his special service to mathematics was that, by study and diligence and the 
exercise of the intellectual power which he possessed, he had been able to 
consolidate some of the conquests made by the great mathematicians, his 
predecessors and contemporaries, and had evolved and excogitated a method 
by which the diligent student of average ability could retread the path 
which had conducted the man of genius to his discoveries. 
