Hi,
Many thanks to aghadowey and Bannerman for your replies.
Other information I have on Alexander Lawrence is -
a. Our Lawrence "coat of arms" is almost identical to the St.Lawrence/Lord Howth "coat of arms" so perhaps
    there is some connection with them. However, our "Coat of arms" appears to be English? See pictures 
    enclosed.
b. He emigrated to South Africa in 1862 and in the book "Aided Immigration from Britain to South Africa 1857-    1867" by Esmè Bull, he is recorded as -
    Permit No: R1520/P1397       Name: LAWRENCE, Alexander       Age: 22       Status: Single                       
    Occupation: Labourer
    Destination: 1 Chapel Street, Capetown. South Africa. (Residence of Brown LAWRENCE)
    From: County Derry, Ireland
    Ship: "John Vanner"
    Arrival Date: March 1862
    Travelling with Alexander was Mary LAWRENCE aged 20, also single. Occupation Housemaid
c. As far as I know, his father was resident in England when he went to South Africa.
 
d. Other Lawrence's that arrived in Cape Town between May 1860 and January 1865 from County Derry     
    were - Brown 19, Hugh Lawrence 29, Eliza 37, James R (Jonathon) 3, David 1 and Catherine Ann 22.