542 



♦ KNOWLEDGE ♦ 



[June 26, 1885. 



A D F H, is equal, by what was shown in dealing with 

 pyramids, to one-third the vohime of a rectangular paral- 

 lelepiped having its base equal in area to A D F H and its 

 height equal to ^' M the height of the cone. 



A. Therefore in the case of a cone on a circular base, 

 having h for its height, and r as the radius of its circular 

 base, the volume of the cone is represented by l^trr'hl 



M. Yea ; and the volume of a cone of height /*, on an 

 elliptical base with semi-axes a, b, is represented by 

 \-7r ahh. 



Fig. 6. 



A. Will it not be convenient to have expressions also for 

 the volume cf frusta of cones ? 



J/. Certainly. Suppose D « B (Fig. 5) to be a fru.stum 

 of a cone, the complete cone being D A B ; the plane d a b 

 parallel to the plane D A B, C in. M perpendicular to both 

 these planes. Then frustum D a B is the difference in 

 volume between the larger cone C D A B and the smaller 

 cone Cdab; hence, from what has been already shown, 

 vol. D rt B = pyr. on area D A B, of height CM — pyr. on 

 area d a b, of ht. C ?». 



If 0M=7(, C 771 = It', and DAB, J ((5 are circles of 



/,' 

 radii DO, do, then if D = r, and do=rj,aac] 



vol. D « B : 



h'- 



-(rV.~~-h') 



= S(^^- 



and if D A B is an IHpse having semi axes A = « and 

 D ::= 6, it is obvi us that 



voh DaB^^(/i3-A'3) 



3/r 



ah' bh' 



tor a = -J— ; and od = -, — 



If we suppose the cone a right circular cone, as in 

 Fig. 6, and that the half-vertical angle D C O is equal 

 to o, then, if C0=/(, C o=h', DO=h tan a, and 

 d o = h' tan a. Hence 



vol. of cone = „ A' tan- a 



vol. DaB = 



r tan - fi 



■{I^-k^) 



If we put 0=k, then 



TT tan- a r ) 



yol. D a B== — "2 — j A-5 - (h - kf | 



K tan- a \ I , 



= — 3 j 3/t {h -]■) +k'-^^ k. 



A. Of what use is this formula's 



M. It is the simplest expression for the volume of a 

 frustum in terms of the height, vertical angle, and the 

 height of the frustum. 



A. What relation does the volume of the frustum D a B 

 bear to the volume of a cylinder having circle K L M (mid- 

 way between DAB and d a b for base), and o O for 

 height ? 



J/. You will find it an easy and pleasant exercise to 

 show that drawing d' o' D' parallel to C D, 



vol. D a B= vol. cyL on base K L M, and of ht. o O 



+ vol. double cone formed by the revolution 

 of triangles D' O o' and d' o o' around o 0. 

 {To be continued.) 



IRISH SEASIDE RESORTS 



FROM AN UNCONVENTIONAL POINT OF VIEW. 

 By Percy Russell. 



(Continued from p. 521.) 



SLIGO, the country of the O'Connors, whose feuds led 

 to the conquest of Ireland, has a very striking and 

 eminently picturesque coast, which is, however, very 

 dangerous indeed for navigation, and excellent and pro- 

 ductive fisheries. 



Next, Mayo, with a coast-line of 250 miles, lofty moun- 

 tains frowning against the Atlantic, with tremendous cliffs, 

 and splendid salmon-fishing in the Moy. Achill Island I 

 have spoken of already, and I may add that, to the tourist 

 seeking really new scenes and fresh and very vivid impres- 

 sions of really romantic coast scenery. Mayo offei-s many 

 inducements — at a favourable season, however, for dread- 

 ful rains prevail at times from the Atlantic side. 



These observations apply with even greater force to 

 Gal way, watered on one boundary by that magnificent river, 

 the Shannon. Southwards are the Slieve-baughta moun- 

 tains, and westward that remarkable group known, curi- 

 ously enough, as the Twelve Pins, about 2,000 feet high. 

 Nothing can well be more romantic than the west portion 

 of these little-frequented and sparsely-peopled regions. 

 The scenery is very wild and romantic ; picturesque glens 

 divide the lofty and rugged hills. The coastline extends 

 for four hundred miles, with remarkable sinuosities, and is 

 fringed with a number of islands. The Connemara moun- 

 tains edge a siugularly wild coast, and well merits its 

 .specific name, Connemara — Cun-na-mar — i.e., boys of the 

 sea. The climate is mild, although rather humid, and the 

 coast is well supplied with excellent fish. Here may be 

 found numerous raths and cromlecJis, monastic ruins, and 

 at least seven of those mysterious round-towers, which still 

 puzzle antiquaries. Clare County extends its coast from 

 the Bay of Galway to the mouth of the Shannon, and, 

 reckoning that fine estuary, has a line of 240 miles. Gene- 

 rally the sea-line itself is high, fantastic, and rocky, there 

 being a great number of small isles and picturesque rocks. At 

 one place the shore runs for full five miles nearly -lOO feet 

 above the sea, and everywhere presenting to it a perpen- 

 dicular face of tremendous clifis. Clare, like Galway, 

 abounds in cromleclts, ruined monasteries, ancient castles, 

 and round-towers. The climate is very much the same. 



Quite as wild, rugged, and picturesque is the coast- 

 line — about 200 miles — of Kerry, which is fringed with a 

 number of islands, and has two very remarkable peninsulas 

 occupied by lofty mountain ranges. Here rise up the 

 famous Macgillicuddy's Reeks, already noted as including 

 the highest mountains in all Ireland, and inland lie the 

 famous lakes of Killarney. The fisheries on the coast are 

 very good, and Dingle Bay, in particular, is remarkable 

 for its splendid and, in some parts, sublime scenery. 



Cork is the mo.st southern and the largest of all the Irish 

 maritime counties. The shores are very bold and uniformly 



