Decembeb 20, 1901.] 



SCIENCE 



975 



But the component of the angular momentum 

 round, the vertical 

 (30) 00co»^-[-CKsin-9 = G'oos^ 



so that 



/Qix clf_G — G: COS ■& 



G—G'z 



dt 



G — G' 1 



G + G^ 1 



' 1 + z' 



+ 



2Ai 1 — z' 2^1 



which gives i> as the sum of two elliptic integrals 

 of the III. kind, their addition into a single in- 

 tegral (Legendre) is shown by (28), (29). 



The reduction to the Weierstrassian form is 

 effected, by putting 



(32) pU- ea. = S Sa = \BP{z Za.), 



where M is a homogeneity factor at our disposal ; 



and now 



■»s^Mds 



(33) nt= j ^^, S=A{s — s^)[s — s,),{s — s,), 



(34) p'^ti^S=iM^Z, 

 nt C ds r" ds 



If V, o, 2 denote the corresponding values of 

 /i, s, S when z=^E, 



(36) pv—piLi = iM%E—z) 



(37) pV—ea. = <7 — Sa = ^M^{E—Za) 



(38) 



ip'v ■ 



/(— 2)=Ji-3. 



(39) 2:1 > -E> ^a > 2 > %, h><^> %> S.> «3. 

 so that 



V =: Wj -f- TpU.^ ; 



and (28) becomes 



^ .11 {*ip' vdn 



(40) TT = - n< + H ^ ' 



k J pv — pii 



with the elliptic integral of the III. kind in the 

 standard form of Weierstrass. 



di~ 

 a constant, H and K coincide, and 



(41) p = GK=OCs\n^ — CKsin^=G^Bin-» 



— Aifj. sin 1^ cos 1?, 



(42) Wgh sin 1? = p^= sin ■» ( G'fi — A^/^^ cos ■»), 

 and dropping the factor sin ■&, 



(43) A^fi^ cos ^ — G'fi-h Wgh = 0. 



A. G. Gbeenhill, 

 Ordnance College, 

 Woolwich, Eng. 



In the steady motion of the top, — =0, 



CURRENT NOTES ON PHYSIOGRAPHY. 

 NEW ORDNANCE MAP OF ENGLAND. 



The 'Colored One inch Map of England,' 

 now in process of publication, marks a great im- 

 provement on both the old and the new series of 

 the earlier ordnance survey inch-to-a mile maps. 

 Relief is shown by brown hachures, drawn with 

 much accuracy of expression, and by red con- 

 tours at intervals of 100 feet. Water is in blue, 

 with blue contours in the sea for every 25 feet of 

 depth near shore ; the chief roads are in ochre, 

 woodland on some of the sheets is green, and 

 culture is black. Most of the. sheets represent 

 quadrangles measuring 18 miles east and west by 

 12 miles north and south ; but for southern Eng- 

 land the sheets frequently include larger areas, 

 according to some system that is not immedi- 

 ately apparent. Some 130 sheets have now 

 been published, the standard sheets costing 

 a shilling each. While looking over them, 

 local geographical features are brought vividly 

 to mind. The Falmouth sheet, Cornwall, in- 

 cludes the typical drowned valleys of Fal and 

 Helston rivers, open to the sea on the east side 

 of the even uplands back of the Lizard, and of 

 Loe river, closed on the more exposed western 

 coast by Portleven sands, one of the few beaches 

 of this ragged shore line. The contours along 

 the valley sides here and on the neighboring 

 Ivybridge and Boscastel sheets are of much 

 smoother curvature than those that follow the 

 coast, thus showing that the shore line in this 

 district of resistant ancient rocks is in that im- 

 mature stage of development when its irregu- 

 larity of detail has become greater than it was 

 in the initial stage. Where the coast consists 

 of weaker Mesozoic rocks, as shown on the 

 Exeter and Sidmouth sheets, a smoother shore 

 line of greater retreat and more mature express 

 sion is found. A little further east, where the 

 Bridport and Weymouth sheets join, the long 

 sweeping curve of Chesil bank is finely dis- 

 played. 



The relations of rivers to their valleys offer 

 some interesting problems. In certain meander- 

 ing valleys the rivers sweep around the valley 

 curves in a most competent fashion, pressing 

 against their outer banks and demanding an 

 increased breadth of meander belt ; the Tor- 

 ridge in Devonshire (Chulmleigh sheet) and the 



