384 Proceedings of the Koyal Society of Edinburgh. [SesS. 
invention of gratings, which have very large dispersion, makes it practicable 
to measure the rotation spectroscopically, and in the years 1887-1889 Duner 
made a beautiful series of observations which determined the law of 
rotation with great accuracy to within 15° of the sun’s poles. The difficulty 
of explaining the phenomena on dynamical grounds made it desirable to 
obtain all the additional information observation could supply. With this 
in mind Dr Halm continued Duner’s observations in the years 1901-1907, 
to determine whether the law of rotation varied with the frequency of Sun 
Spots, which have a well-determined but unexplained period of eleven years. 
He followed Duner in basing his determination on the comparison of two 
lines of solar origin with two neighbouring lines caused by absorption in 
the earth’s atmosphere. He differed from him in the employment of a fixed 
spectroscope fed by a heliostat in place of an equatorial mounting, and still 
further simplified the conditions of the observation by the use of a helio- 
meter which enabled him to compare simultaneously opposite extremities 
of any diameter of the sun. By these means considerable increase of 
accuracy was obtained. The accuracy of the observations was illustrated 
in an interesting manner by deducing the speed of the earth’s rotation from 
a comparison of morning and evening observations, and the eccentricity of 
the earth’s orbit from the observations of spring and autumn. 
As regards the rotation of the sun, Dr Halm found the same equatorial 
velocity as Duner, but a greater diminution towards the poles. Another 
important result brought to light by these researches is a small displace- 
ment towards the red, irrespective of rotation, of the lines of the spectrum 
near the sun’s limb, as compared with their positions in the spectrum of the 
centre of the disc. This displacement, which appears to vary with the Sun 
Spot cycle, is accounted for by Dr Halm as the result of pressure in the 
stratum of the sun which the light traverses. 
Among Dr Halm’s other astronomical papers reference may be made to 
one on “Line and Band Spectra,” in which he shows that these may be 
included in a single formula ; and to one on “ Temporary Stars,” where he 
develops in detail a hypothesis of Professor Seeliger’s, that a new star 
becomes visible when a dark body impinges upon and penetrates into a 
mass of nebular material. 
The memoir “ On a Group of Linear Differential Equations of the Second 
Order, including Professor Chrystal’s Seiche-equations,” is a valuable con- 
tribution to the solution of a type of equations which play an important 
role in mathematical physics. It enriches the theory by investigating the 
solutions for other than integral values of one of the parameters of the 
equation, and throws a new light on the theoretical importance of the 
