392 SECTIONAL TRANSACTIONS.— A, A*. 



to the arrival of long Rayleigh waves, the latter to long Love waves. An 

 analysis of a number of earthquakes listed in the I.S.S. shows a marked 

 separation of the L readings into two groups, with travel-times clustering 

 round the above values. The relative proportions of the two types suggest 

 intrinsic differences in the dislocations that initiate the waves. Differences 

 are shown for earthquakes in the same region, so that the effect cannot be 

 attributed entirely to the distribution of stations or of land and sea. To 

 ascertain the dependence on intensity of shock a reliable measure of in- 

 tensity would be needed. 



Prof. J. D. Bernal, F.R.S. — Crystallographic relations of seismology 

 (3-I0)- 

 A possible explanation of the 20° discontinuity is to be found in the 

 hypothetical existence at lower levels of the earth's crust of a denser and 

 more elastic crystal form of olivine (MgFe)2Si04. Such a form has never 

 been observed owing to the impossibility of producing sufficiently high 

 pressure, but it might be expected that the condition of silicates at high 

 pressure should be shown by germanates at ordinary pressures, owing to 

 the greater size of the germanium ion. Magnesium germanate has been 

 observed to exist both in olivine structure and in the cubic spinel type of 

 structure. A calculation based on this structure would seem to be able 

 to account in a rough quantitative way for the properties required for the 

 substances at low levels. 



DEPARTMENT OF MATHEMATICS (A*). 



Thursday, August 18. 



Symposium on Newtonian root evaluations (lo.o). 

 Chairman : Dr. J. Wishart. 



Prof. A. OsTROWSKi. — On Newton's method of approximation (lo.o). 



To compare the amount of work in Newton's method of approximation 

 for a root ^ of the equation /(ar) = o with the effect of the method, a unit of 

 the calculatory work is introduced — a Horner, that is the work of calculating 

 the value of P^^z), this unit being assumed as sensibly independent of 

 V and the number of digits in z. 



On the other hand, the rate of the approximation of ^ by y is measured 

 by the order of | y — ^ | , if the initial value x^ tends to K, the order of | Xo — C| 

 being assumed as i. 



Newton's method gives with 6 Horners an approximation of the order 8. 

 A modification of Newton's method is proposed allowing to obtain with 

 6 Horners an approximation of the order 16. It consists in using alternately 

 the two formulae : 



""' ""' fixoY ""'-'^^ 2f{x,)-f{x,)- 



Prof. E. H. 'NEVihhE.— Computational labour in modifications of 

 Newton's method of approximation (10.30). 



The processes of interpolation have been developed on the assumption 

 that the arguments for which they are to be used may be anywhere in the 



