1911-12.] Obituary Notices. 495 
literature of the subject by bringing out his Introduction to Algebra for 
the use of secondary schools and colleges. 
Although he did not for many years take up any experimental work, 
he was always ready when occasion offered to advise and help others 
engaged in such work. For example, he took a keen interest in the Ben 
Nevis Observatory, and devised special forms of hygrometer and ane- 
mometer for use in these altitudes, especially during the winter season. 
He also turned his attention to the invention of an electrical method for 
reversing deep-sea thermometers. 
In 1883, through the initiative of the late Mr Yule Fraser, at that time 
mathematical master in George Watson’s College, the Edinburgh Mathe- 
matical Society was founded. As its first secretary I was brought into 
still closer touch with Chrystal, who took a warm interest in its welfare 
from the first. In these early years he contributed a number of notes on 
mathematical subjects, and greatly encouraged the members by his presence 
at the meetings. His paper on certain inverse roulette problems contains 
elegant solutions of particular cases of the general problem : Given the body 
centrode and the roulette for one point of a plane figure moving in its 
plane, to find the space centrode. The most elaborate paper which he 
communicated to the Mathematical Society was entitled “ On the Theory of 
Refraction of approximately Axial Pencils of Light through a Series of 
Lenses, more especially with regard to Photographic Doublets and Triplets.” 
He was led to take up the subject in connection with his own photographic 
work. The theory is worked out in a very simple form, and instructions 
are given the student of photography how most easily to determine the 
constants of his system of lenses. 
Between 1891 and 1896 Chrystal communicated three fairly elaborate 
papers on differential equations. There was a rumour at one time that he 
purposed writing a book on this subject ; whether this was so or not, the 
character of these papers shows that he had given careful consideration to 
the logical foundations of the theory of certain parts. For example, in the 
first paper he proves the defective nature of Lagrange’s demonstration 
of the rule for the solution of the partial differential equation of the 
first order, and supplies a demonstration free from the fallacy which had 
been generally current since Lagrange’s days. Similarly, in the second 
paper he aims at establishing rigorously a fundamental theorem regarding 
the equivalence of systems of ordinary linear differential equations. This 
leads to a systematic way for solving systems of this kind without the 
introduction of superfluous arbitrary constants ; and the paper ends with 
illustrations of the practical use of the method. In the third paper, that 
