490 TRAIVSACTIONS OF SECTION A. 



equation whicli make two of the three quantities P, Q, E, vanish. There remain 

 those integrals f = (if any) which make one of the three quantities P, Q, R 

 vanish. 



This last possibility has another hearing. During the discussion we have 

 assumed that u and v are regular functions of a; y, z in the vicinity of values 

 which satisfy ^1^ = 0; but when such values make either r or Q or R vanish, the 

 equations 



dx dy dz 

 P" = Q ~ R 



do not necessarily possess regular integrals. Further, Cauchy's existence-theorem 

 is silent as to the possession of regular integrals for initial values, which constitute 

 a branch-place or a singularity of P or Q or R. 



Summing up the results, we see that there may be leakage in the compre- 

 hensiveness of the general integral f (u, v) = 0, so far as concerns a special integral 

 y\r=0,when values of the variables satisfying ^ = Q are such as to malie either 

 P or Q or R vanish, or are such as to constitute any deviation from regularity 

 in u or in v, the quantities out of which the general integral is composed. 



Similar remarks apply to linear equations in more than two independent 

 variables, to equations of the first order that are not linear, to equations of the 

 second order having intermediate general integrals, and to equations of order 

 higher than the second. Sufficient has been said to indicate that special integrals 

 can exist which are not included in the general integral, and to hint that the 

 customary classification of integrals of partial equations of various orders requires 

 revision. 



Department of Astronomy and Cosmical Physics. 



The following Reports and Papers were read :— 



1. Report on the Invesligation of the Upper Atmosphere hy means 

 of Kites. —See Reports, p. 138. 



3. Reiiort of the Seismological Committee.— Hne Reports, p. 92. 



3 The Irreffular Motions of the Earth's Pole : a Preliminary Graphical 

 Analysis of their Cause. By Major E. H. Hills, C.M.G., and 

 Professor J. Larmob, Sec.R.S. 



4. Ifote on the Effects of Tremors on Astronomical Observations. 

 By Professor H. H. Turner, D.Sc, F.R.S. 



In his last report the Astronomer Royal drew attention to the fact that the 

 generating station erected by the London County Council half a mile to the north 

 of the Observatory at Greenwich produced disturbing effects on the observations — 

 among others tremors sufficient to shake the Observatory. The effect of such 

 tremors has been little studied recently, but half a century ago, when railways 

 were first being brought near the Observatory, a number of experiments on tremor 

 were made, not only at Greenwich, but elsewhere. An extreme case of dis- 

 turbance occurred at the 'Armagh Observatory,' where the Ulster railroad 

 approached within 700 yards, and Dr. Robinson wrote a paper in the ' Proceed- 

 ings of the Royal Irish Academy ' (vol. v. p. 287) calling attention to several 

 ways in which tremors disturhed the observations. The best known is, of 

 course, the shaking of the mercury trough with which observations are made by 



