Jan. 7, 1886] 



NATURE 



could create such a School of Mathematics as might go 

 some way at least to revive the old scientific renown of 

 Oxford, and to light such a candle in England as, with 

 God's grace, should never be put out.' 



TABLES OF SINGULARITIES AND FORMULA 

 REFERRED TO IN THE PRECEDING LECTURE 



Chart i. 



POJXTS OF ilAXnWM AND MINIMUM CURTATUIiE 



Chart 2. — PROTO^roRPHS 

 Binariniils ^ Rccifro:anls 



,u - b"- 



a"d - 3(7fc + 2p 



a-f+ Serie + 2acl+ Sl'-d - kbc 



rt»-6i/+I5(-f- 10,/ = 



9nV-45(T(5c + 4o^^ 



5<jV-3S<7W+7rtr- + 35^V 



45«y- 420a-/v - 42aVrf+ 1 1 zoali'd 



-TjX^ahc"-- ii2o/;V 



a'-g- I2abf- 450(7C(?+ 'jgife 



+ $'&?>aie" - 2T]2k(/+ Iijz^c^ 



Chart 3. 

 Xo. I. a 

 No. 2. s-Jc- 5*° 

 No. 3. 9«V-45rtfe + 4oi^ 

 No. 4. 45flV - 450rj%- + I92(7V' + 4on,7/-V+ i6~,ab"c- - 40oiV 





idt 



+ M 



*" .' \/k(I - I5<2+ 15/4 _ ^6^ + X(lt- 10^3 + 3/") "*" '' 



V -la-ti, + lo.iifSf + (i5a(r+io5=)Srf+(2ia</+35^,-)5, 

 + (28af + s6i5a'+35c'')5^+ . . . 



I I have pui-posely confined myself in my lecture to reciprocants, indica- 

 lives of properties of plane curves, but had in view to extend the theory to 

 ihe case of higher dimensions in space leading to reciprocants involving the 

 differential derivatives of any number of variables ji*, 2, . . . M. Halphen, with 

 whom I have had the great advantage of being in communication during my 

 stay in Paris, has anticipated me in this part of my plan, and has found that 

 the same method which I have used to obtain the Annihilator /'applied to a 

 system of variables lead^ to an Annihilator of very similar form to /', and at 

 my request will publish his results in a forthcoming number of the Coinjtics 

 rettdus. Thus the dominion of reciprocants is already extended over the 

 whole range of forms unlimited in their own number as well as in that of the 

 variables which they contain 



Chart 4.— Coefficients of Annihilator V 



35 

 126 

 210 



Cnih 



[a) 



Chart 5.— Reciprocant Transformations 



Chrysalis 

 d-<p d'(t> 



dx^ 

 ,P^ 

 dxdy 

 dip 

 dx 



dxdy 

 d'lp 



d^ 

 dy 



d^ 

 dx- 

 d-* 

 dxdy 

 d-* 

 dxdz 



Imago 



dxdy 

 d'i 

 -df 

 d-<i> 

 dydz 



d-^ 

 dxdz 

 d-* 

 dydz 

 d-^ 

 d'J 



^— {Si 



dy^ 



fi^YU = o. 

 \dxdj I 



dy ih^ 

 dx dxi 



' Z\ is the Schwarzian, otherwise written ili-^—' 



Chart 6. — The H Recipkocantive Protomorph 



U 

 6sa*h 



- 975aV.^ 



- ggoa^cf 

 + 620oa'-d-/ 

 + /^Sgoa-ice 



- is^oai'e 



- 2J3oa-6ii" 

 + Ji6ia^c-d 

 + 3oSoriiW 

 -24255a/^f' 

 + 254ioi';V2 



ir 



I20a'^i/ 



- 200a-i-/ 



- igSa'de 



- I45a^(''« 

 + loooai^" 



7/ie Vermicular Operator 

 \aS/. +ii.iSc + vcS^ + TrdSe + . . . 



i + l 



Examples 



' '~~c~"ii:j- I a04 + »(if +f5rf + a'5, + . . . 



+ I36S«W- a5* + 2A8. + 3fSrf+4«'5,+ . . . 



-■jTjac-d 3a5j+8M, + i5fBrf+24a'S,+ . . . 



222()0al>-cit\ ■' 

 + 2485<7&s 



+ io5/'V- 1 «S" b*d doesnot appear in either 



6^ or IV. 



H*hU+MW 



A and M are arbitrary numbers. 



New College, Oxford, January 6 



THE GEOLOGY OF MALAYSIA, SOUTHERN 

 CHINA, &-C. 



THERE is a remarkable uniforniity in the geology of 

 a very large portion of Southern Asia and its 

 dependent islands, especially from the Malay peninsula, 

 as far east as the Philippines, and as far north as the 

 Chinese continent. In the Malayan peninsula we have 

 an elevated granitic axis. At the base of this there are 

 Pateozoic schists and slates. Above these in a few 

 places there are limestones in detached weathered 

 masses. This limestone is often crystalline, white, blue, 

 and black. In a few cases there are traces of stratifica- 

 tion, but no fossils. 



In a recent journey through Pahang I found precisely 

 the same formations on the eastern side of the peninsula, 

 with only this addition, that there is a belt of trachytic 

 rocks of modern origin forming detached hills between 

 the main range and the sea. 



In Sumatra I learn that there are the same formations 

 from the granite upwards. I cannot confiriTi this from 

 personal observation, as I have travelled very little in the 

 island. The mountain axis is far from the Straits of 

 Malacca, and difficult of access. As far as I can judge 

 from the geology of such large islands as Bilitou, Bintang, 

 and Banca, the mountains are probably granitic and 

 stanniferous. 



Proceeding eastward and northerly, detached granite 

 islands are met with. They are thickly strewn through 

 the intermediate ocean. Those I have seen, such as the 

 north and south Natunas, and other similar outliers, on 

 vovages between Java, Singapore, Borneo, China, Cochin 



