262 Royal Society. 
tained results which —— the ratios of the series of measure- 
ments rici obtain 
'The second case of "discoid aec in which m=k and 2—0, is by 
far the te commoner, as to it belong all genera of discoidal mollusks, 
with the few exceptions noticed iau The case m > kis one which 
cannot occur, as then the outer whorl must necessarily crush the 
inner, and then the penang figure could not retain its | 
trieal identity while enlarging ; hence we find no examples of it in 
discoid shells a 
have placed i in this second case some instances in which the 1 
ratio of slipping or translation on the axis is not easily measured, 
and virtually amounted to nothing. 
The following Table of examples illustrate case No. 2:— 
Species. Mim EEUU Width of whorls in decimals of an inch. | 
I 0:075 |0'75 ý 
Mahotis indir i.e 10 | Ellipse .... { 0-05 (05 í d 
| 0:15 |r5 . 
a E xpo | os 0°02 os rö 
= s reret O08 10:98 [27 
Sulculus (Haliotis) parvus ........ 6 War Lus 0°03 (017 |! 
Padollus (Haliotis) excavatus...... 4°2 | Ellipse ...... 9-06 (0°25 |1*1 
Natiea canrena 3 i. req of j 0-025 |0-075/0°25 0°76 
Nautilus pompilius 3 1 Segme t 9€ a lo-anós odo [ros 
DOptd uices Tirai. } 1 
Dolium zonatum ................ a4] ul. 0:119 |0*25 |07525 
Solaropsis pelli pentis ........ 2 E d 9:023 |0'047|U-086 0'17 |0'34 
p X Segment o s 5 d ^ 
3 i circle .... J [V 02 ul emo 
Eucmphalus pentangulatus,....... Sree Ue 0 
MK 21e 1°75 | Rhomboid 0:07 {0712 |0"2 5 10-65 
Architectonica trochleare .......... 1:62 ee eee. /0°046 |0:075/0175/072 10:325 0:55 
143, Triangle ....|0-02 0°03 |0*05 |0-072 0-09 |0:12 |0"17|0"25 
Conus literatus 1*4 a ....]0703 [0°04 [0°05 0-86 0:125 07176 0°25 
onus virgo 1:25 A ....10708 [OI  |0:105/0:16 
Planorbis, sp 1:38 fakes 003 |0042 IT 01 [0°15 |08 
discoidal patoons, m; thirdly, the helicoidal coefficient, x. “Upon 
such 
the whorls c embrace each other, and Hes rs Rees 
that of an elongated cone, as in the genera Turritella, Cerithium, 
Acus, &c. Sometimes n exceeds m; and in this case the re ting 
form is an open spiral as in V ermetus, or a rapidly descending series 
of whorls. A third possible case is that in which z is less than 
m, and the resulting figure is globular; but of this case, though a 
possible one, I have not as yet su in o mod an example. 
The following cases illustrate the formula n> m 
